Sach excel 2013 for social sciences statistics a guide to solving practical problems by thomas j quirk english

Mean

The mean, often referred to as the "arithmetic average," represents the central value of a set of scores When my daughter, a fifth grader, came home feeling upset about not understanding averages, I took the opportunity to explain how to calculate the mean based on her school book.

Jennifer insisted on the seriousness of the situation as she urged me to add up all the scores and divide by the total count Her expression made it clear that she believed I was joking, but I was genuinely trying to help her understand the concept.

“See these numbers in your book; add them up What is the answer?” (She did that).

“Now, how many numbers do you have?” (She answered that question).

“Then, take the number you got when you added up the numbers, and divide that number by the number of numbers that you have.”

By applying the same reasoning, you will discover the correct answer effortlessly, as Excel automates all the necessary steps for you.

We will call this average of the scores the “mean” which we will symbolize as:X, and we will pronounce it as: “Xbar.” © Springer International Publishing Switzerland 2015

T.J Quirk, Excel 2013 for Social Sciences Statistics, Excel for Statistics,

The formula for finding the mean with you calculator looks like this:

The Greek letter sigma (Σ) represents the concept of "sum," instructing you to total all the values indicated by the letter X and then divide that total by n, which denotes the count of the numbers involved.

A political scientist conducted a survey to gauge registered voters' attitudes on various issues, including the belief that wealthy individuals should be taxed at significantly higher rates than those with lower incomes The survey utilized a 7-point rating scale to assess respondents' opinions on this statement.

1ẳstrongly disagree, and 7ẳstrongly agree.

Suppose that you had these six ratings on this one item:

To calculate the mean of a set of scores, sum all the scores together and divide by the total number of scores For example, adding the scores yields 25, and dividing by 6 results in a mean of approximately 4.17, which is near the midpoint of a 7-point scale.

Standard Deviation

Standard deviation measures the proximity of scores to the mean, indicating how closely they cluster A small standard deviation signifies that the scores are tightly grouped around the mean, while a large standard deviation indicates a wider dispersion of scores The standard deviation is represented by the formula STDEV, symbolized by the letter S.

The formula look complicated, but what it asks you to do is this:

1 Subtract the mean from each scoreXX

2 Then, square the resulting number to make it a positive number.

3 Then, add up these squared numbers to get a total score.

2 1 Sample Size, Mean, Standard Deviation, and Standard Error of the Mean

4 Then, take this total score and divide it by n1 (where n stands for the number of numbers that you have).

5 The final step is to take the square root of the number you found in step 4.

This article focuses on calculating the standard deviation using Excel rather than manual computation with a calculator For demonstration, we will analyze a set of six numbers previously mentioned, and when we apply the STDEV function in Excel, we find that the standard deviation, S, is 1.47.

Standard Error of the Mean

The formula for the standard error of the mean(s.e., which we will use S X to symbolize) is: s:e:ẳS X ẳ S

To calculate the standard error (s.e.), divide the standard deviation (STDEV) by the square root of n, where n represents the total number of values in your data set For instance, in the example provided, the s.e is 0.60, which you can verify using a calculator.

If you want to learn more about the standard deviation and the standard error of the mean, see Weiers (2011) and Neuman (2000).

In this article, we will explore how to utilize Excel to calculate essential statistical metrics, including sample size, mean, standard deviation, and standard error of the mean, based on a geometry test administered to 9th graders at the end of the first term, with a total of 50 points possible The hypothetical data is illustrated in Fig.1.1.

Fig 1.1 Worksheet Data for a Geometry Test

1.3 Standard Error of the Mean 3

Sample Size, Mean, Standard Deviation, and Standard

Using the Fill/Series/Columns Commands

Objective: To add the student numbers 2–8 in a column underneath student #1

Home (top left of screen)

Fill (top right of screen: click on the down arrow; see Fig.1.2)

Fig 1.2 Home/Fill/Series commands

The student numbers should be identified as 1–8, with 8 in cell A11.

Now, enter the Geometry Test Scores in cells B4:B11.

Since your computer screen shows the information in a format that does not look professional, you need to learn how to “widen the column width” and how to

“center the information” in a group of cells Here is how you can do those two steps:

Changing the Width of a Column

Objective: To make a column width wider so that all of the information fits inside that column

If you look at your computer screen, you can see that Column B is not wide enough so that all of the information fits inside this column To make Column

Click on the letter, B, at the top of your computer screen

Place your mouse pointer at the far right corner of B until you create a “cross sign” on that corner

Left-click on your mouse, hold it down, and move this corner to the right until it is

“wide enough to fit all of the data”

To adjust the column width, simply release the mouse button (refer to Fig 1.4) Additionally, Fig 1.3 illustrates the dialogue box for the Fill/Series/Columns/Step Value/Stop Value commands This section also covers essential statistical concepts, including sample size, mean, standard deviation, and the standard error of the mean.

Then, click on any empty cell (i.e., any blank cell) to “deselect” column B so that it is no longer a darker color on your screen.

When you widen a column, you will make all of the cells in all of the rows of this column that same width.

Now, let’s go through the steps to center the information in both Column A andColumn B.

Centering Information in a Range of Cells

Objective: To center the information in a group of cells

In order to make the information in the cells look “more professional,” you can center the information using the following steps:

Left-click your mouse on A3 and drag it to the right and down to highlight cells A3: B11 so that these cells appear in a darker color

At the top of your computer screen, you will find a series of lines that are uniformly centered in width under the "Alignment" settings, which is the second icon located at the bottom left of the Alignment box (refer to Fig 1.5).

Fig 1.4 Example of How to Widen the Column Width

Click on this icon to center the information in the selected cells (see Fig.1.6) Fig 1.5 Example of How to Center Information Within Cells

Centering Information in the Cells

1.4 Sample Size, Mean, Standard Deviation, and Standard Error of the Mean 7

To simplify referencing the Geometry Test Scores in your formulas, it's beneficial to name the data range instead of recalling specific cell locations like B4:B11 For instance, you can label this group of cells as "Geometry," but feel free to choose any name that suits your preference.

Naming a Range of Cells

Objective: To name the range of data for the test scores with the name:

Highlight cells B4:B11 by left-clicking your mouse on B4 and dragging it down to B11

Formulas (top left of your screen)

Define Name (top center of your screen)

Geometry (type this name in the top box; see Fig.1.7)

Then, click on any cell of your spreadsheet that does not have any information in it (i.e., it is an “empty cell”) to deselect cells B4:B11

Fig 1.7 Dialogue box for “naming a range of cells” with the name: Geometry

Now, add the following terms to your spreadsheet:

When using formulas in Excel, it is essential to start each formula with an equal sign (=) to indicate to the program that you are inputting a formula This simple step ensures that Excel recognizes your intention and processes the calculation correctly.

Finding the Sample Size Using

Objective: To find the sample size (n) for these data using the ẳCOUNT function

This command should insert the number 8 into cell F6 since there are eight students in this class.

Fig 1.8 Example of Entering the Sample Size, Mean, STDEV, and s.e Labels

Finding the Mean Score Using

Objective: To find the mean Geometry figure using theẳAVERAGE function

This command should insert the number 23.125 into cell F9.

Finding the Standard Deviation Using

Objective: To find the standard deviation (STDEV) using theẳSTDEV function

This command should insert the number 14.02485 into cell F12.

Finding the Standard Error of the Mean

Objective: To find the standard error of the mean using a formula for these eight data points

This command should insert the number 4.958533 into cell F15 (see Fig.1.9).

It is crucial to verify that all figures in your spreadsheet are accurately placed in their respective cells, as any discrepancies will lead to incorrect formula calculations.

1.4.8.1 Formatting Numbers in Number Format (Two Decimal Places)

Objective: To convert the mean, STDEV, and s.e to two decimal places

Home (top left of screen)

To decrease the number of decimal places displayed in your document, locate the "Number" section at the top center of your screen Then, move your mouse pointer to the bottom right corner of the decimal display, specifically over the 00.0 area, until you see the option labeled "Decrease Decimal."

Fig 1.9 Example of Using Excel Formulas for Sample Size, Mean, STDEV, and s.e.

Click on this icononceand notice that the cells F9:F15 are now all in just two decimal places (see Fig.1.11)

Fig 1.10 Using the “Decrease Decimal Icon” to convert Numbers to Fewer Decimal Places

Fig 1.11 Example of Converting Numbers to Two Decimal Places

Now, click on any “empty cell” on your spreadsheet to deselect cells F9:F15.

Saving a Spreadsheet

Objective: To save this spreadsheet with the name: Geometry3

To save your spreadsheet for future access, first determine the location where you want to store it You have several options, including saving it on your computer's hard drive, a CD, or a flash drive If you're unsure how to save it on your computer, seek assistance.

To save a file, simply scroll through the left sidebar to choose your desired location, such as "My Documents," and click on it to confirm your selection.

File name: Geometry3 (enter this name to the right of File name; see Fig.1.12)

Fig 1.12 Dialogue Box of Saving an Excel Workbook File as “Geometry3” in My Documents location

Important note: Be very careful to save your Excel file spreadsheet every few minutes so that you do not lose your information!

Printing a Spreadsheet

Objective: To print the spreadsheet

Use the following procedure when printing any spreadsheet.

Print Active Sheets (see Fig.1.13)

Fig 1.13 Example of How to Print an Excel Worksheet Using the File/Print/Print Active Sheets Commands

Print (top of your screen)

The final spreadsheet is given in Fig1.14

Before concluding this chapter, let's practice formatting figures in a spreadsheet through two examples: first, by displaying dollar amounts with two decimal places, and second, by formatting other figures to show three decimal places.

Close your spreadsheet by: File/Close, and open a blank Excel spreadsheet by using File/New/Blank Workbook (in the top left of your screen).

Formatting Numbers in Currency Format (Two Decimal Places)

Objective: To change the format of figures to dollar format with two decimal places

Highlight cells A4:A6 by left-clicking your mouse on A4 and dragging it down so that these three cells are highlighted in a darker color

Fig 1.14 Final Result of Printing an Excel spreadsheet

1.7 Formatting Numbers in Currency Format (Two Decimal Places) 15

Number (top center of screen: click on the down arrow on the right; see Fig.1.15)

Decimal places: 2 (then see Fig.1.16)

Fig 1.15 Dialogue Box for Number Format Choices

Fig 1.16 Dialogue Box for Currency (two decimal places) Format for Numbers

The three cells should have a dollar sign in them and be in two decimal places.Next, let’s practice formatting figures in number format, three decimal places.

Formatting Numbers in Number Format (Three Decimal Places)

Objective: To format figures in number format, three decimal places

Highlight cells A4:A6 on your computer screen

Number (click on the down arrow on the right)

At the right of the box, change two decimal places to three decimal places by clicking on the “up arrow” once

Ensure that the three figures are formatted as numbers with three decimal places Next, click on any empty cell to deselect the range A4:A6 Finally, close the file by navigating to File > Close and selecting "Don't Save," as saving is unnecessary for this practice exercise.

You can use these same commands to format a range of cells in percentage format (and many other formats) to whatever number of decimal places you want to specify.

End-of-Chapter Practice Problems

A political science professor at a major U.S university conducted a survey to assess undergraduate attitudes towards U.S.-Chinese relations After pretesting the survey with a small group of students, she analyzed the results from item #8, which produced hypothetical data illustrated in Fig 1.17.

1.9 End-of-Chapter Practice Problems 17

To analyze the data effectively, utilize Excel to calculate the sample size, mean, standard deviation, and standard error of the mean Ensure to label each result clearly and round the mean, standard deviation, and standard error to two decimal places, applying the appropriate number format for these values.

(b) Print the result on a separate page.

(c) Save the file as: China7

The Human Resources department conducted a "Morale Survey" targeting all middle-level managers, and I have been tasked with summarizing the findings To evaluate my Excel skills, I focused on item #21 from the survey data presented in Fig 1.18 The analysis of this specific item will provide insights into the overall morale among middle-level management within the company.

Fig 1.18 Worksheet Data for Chap 1: Practice Problem #2

1.9 End-of-Chapter Practice Problems 19

To analyze the ratings, create a table in Excel and calculate the sample size, mean, standard deviation, and standard error of the mean for the data Ensure to label each result clearly and round the mean, standard deviation, and standard error to two decimal places using the number format feature in Excel.

(c) Save the file as: MORALE4

At Deer Creek Elementary School in Bailey, Colorado, a 5th grade science teacher utilizes a basic geology textbook, which generally takes eight class days to cover each chapter Following Chapter 8, the teacher administers a 15-item true-false quiz to assess student understanding, with the results illustrated in Figure 1.19.

To analyze the given data, create a table in Excel and calculate the sample size, mean, standard deviation, and standard error of the mean Ensure to label each result clearly and round the mean, standard deviation, and standard error to three decimal places using the number format feature in Excel.

(c) Save the file as: SCIENCE8

Fig 1.19 Worksheet Data for Chap 1: Practice

Neuman, W.L Social Research Methods: Qualitative and Quantitative Approaches (4 th ed.). Boston, MA: Allyn and Bacon, 2000.

Weiers, R.M Introduction to Business Statistics (7 th ed.) Mason, OH: South-Western Cengage Learning, 2011.

Suppose that a local school superintendent asked you to take a random sample of

5 of an elementary school’s 32 teachers using Excel so that you could interview these five teachers about their job satisfaction at their school.

To conduct a random sample, it is essential to establish a "sampling frame," which is a comprehensive list of individuals from whom the sample will be drawn In this case, the sampling frame begins with the identification code (ID) assigned to each teacher, starting with the first teacher labeled as ID 1, followed by ID 2 for the second teacher, and continuing sequentially up to ID 32 for the last teacher in the list of 32 educators at the school.

Since this school has 32 teachers, your sampling frame would go from 1 to

32 with each teacher having a unique ID number.

We will first create the frame numbers as follows in a new Excel worksheet:

Creating Frame Numbers for Generating

Objective: To create the frame numbers for generating random numbers

To create frame numbers in column A using the Home/Fill commands, follow these steps: Start by selecting cell A1 and input the number 1 Next, drag the fill handle down to cell A35, which should automatically populate the cells with consecutive numbers up to 32 Ensure that cell A35 displays the number 32, completing the sequence from 1 to 32.

Click on cell A4 to select this cell

Fill (then click on the “down arrow” next to this command and select)

Then, save this file as: Random29 You should obtain the result in Fig.2.3.

Fig 2.1 Dialogue Box for Fill/Series Commands

Fig 2.2 Dialogue Box for Fill/Series/Columns/Step value/Stop value Commands

Now, create a column next to these frame numbers in this manner:

To format your spreadsheet, use the Home/Fill command to populate frame numbers starting from cell B4 to B35 Ensure that columns A and B are widened to accommodate all content, and center the information within both columns Your final layout should resemble the example shown in Fig 2.4.

Fig 2.3 Frame Numbers from 1 to 32

2.1 Creating Frame Numbers for Generating Random Numbers 25

Save this file as: Random30

To ensure accuracy in your spreadsheet, you may have duplicated the information in both Column A and Column B This duplication is intentional, allowing you to verify that you have exactly 32 frame numbers before sorting them into a random sequence.

Now, let’s add a random number to each of the duplicate frame numbers as follows:

Creating Random Numbers in an Excel Worksheet

C3: RANDOM NO (then widen columns A, B, C so that their labels fit inside the columns; then center the information in A3:C35)

Next, hit the Enter key to add a random number to cell C4.

To use the RAND() function correctly, ensure that you include both an open parenthesis and a closed parenthesis after RAND This command generates a random number in the cell where it is applied by referencing the cells to its left.

To add a random number to all 32 ID frame numbers, position your mouse pointer over cell C4 and drag it to the bottom right corner until a “plus sign” appears Then, left-click and drag down to cell C35.

Then, click on any empty cell to deselect C4:C35 to remove the dark color highlighting these cells.

Now, let’s sort these duplicate frame numbers into a random sequence:

Fig 2.5 Example of Random Numbers

2.2 Creating Random Numbers in an Excel Worksheet 27

Sorting Frame Numbers into a Random Sequence

Objective: To sort the duplicate frame numbers into a random sequence

Highlight cells B3:C35 (include the labels at the top of columns B and C) Data (top of screen)

Sort (click on this word at the top center of your screen; see Fig.2.6)

Sort by: RANDOM NO (click on the down arrow)

Smallest to Largest (see Fig.2.7)

Fig 2.6 Dialogue Box for Data/Sort Commands

Click on any empty cell to deselect B3:C35.

These steps will produce Fig.2.8with the DUPLICATE FRAME NUMBERS sorted into a random order:

Fig 2.7 Dialogue Box for Data/Sort/RANDOM NO./Smallest to Largest Commands

2.3 Sorting Frame Numbers into a Random Sequence 29

Important note: Because Excel randomly assigns these random numbers, your

Excel commands will produce a different sequence of random numbers from everyone else who reads this book!

Because your objective at the beginning of this chapter was to select randomly

5 of this school’s 32 teachers for a personal interview, you now can do that by selecting the first five ID numbers in DUPLICATE FRAME NO column after the sort.

In this chapter, we will select five unique teacher IDs for interviews, which will differ from the random numbers previously generated, as illustrated in Fig 2.9.

Each time you use the RAND() function in Excel, it generates a unique set of five ID numbers, which will differ from those shown in Fig 2.9.

If you want to learn more about the purpose of taking a random sample in social science research, see Frankfort-Nachmiaset al (2008).

Before concluding this chapter, it's essential to understand how to print a file so that all its information fits neatly onto a single page, avoiding any overflow onto additional pages.

Printing an Excel File So That All of the Information

Objective: To print a file so that all of the information fits onto one page

The practice problems at the end of this chapter involve sorting random numbers from files containing 63 children, 114 counties in Missouri, and 76 key accounts To ensure these files fit on a single printed page, it is essential to format them appropriately, as they may be too large to print otherwise.

Let’s create a situation where the file does not fit onto one printed page unless you format it first to do that.

Go back to the file you just created, Random 33, and enter the name:Jennifer into cell: A50.

Printing this file will cause the name "Jennifer" to appear on a second page due to it exceeding the page margins in its current format.

To ensure that all information, including the name Jennifer, fits onto a single page when printing, you need to adjust the page format by following specific steps.

Page Layout (top left of the computer screen)

(Notice the “Scale to Fit” section in the center of your screen; see Fig.2.10)2.4 Printing an Excel File So That All of the Information Fits onto One Page 31

Hit the down arrow to the right of 100 %once to reduce the size of the page to 95 %

In Fig 2.11, Jennifer's name appears on the second page of your screen, positioned below the horizontal dotted line that indicates the file's outline dimensions for printing.

To reduce the size of your worksheet to 90% of its normal size, simply press the down arrow on the right once more to repeat the "scale change steps." As shown in Fig 2.12, the dotted lines on your screen now appear below Jennifer's name, signifying that all content, including her name, is formatted to fit onto a single printed page.

Fig 2.11 Example of Scale Reduced to 95 % with “Jennifer” to be Printed on a Second Page2.4 Printing an Excel File So That All of the Information Fits onto One Page 33

Save the file as: Random46

Print the file Does it all fit onto one page? It should (see Fig.2.13).

Fig 2.12 Example of Scale Reduced to 90 % with “Jennifer” to be printed on the first page (note the dotted line below Jennifer on your screen)

End-of-Chapter Practice Problems

Howell et al (2000) investigated the impact of private school education on underprivileged children by implementing programs in three U.S cities These programs provided partial scholarship vouchers to allow low-income public school students to attend private schools Due to high demand, more children applied for the scholarships than could be accommodated.

Spreadsheet of 90 % Scale to Fit

In the end-of-chapter practice problems, children were randomly chosen to participate in a voucher program If you were tasked with this selection process, you could enhance your Excel skills by randomly selecting 15 children for the program.

63 children who applied for this program before you did the actual random selection of all of the students who applied for the program.

(a) Set up a spreadsheet of frame numbers for these children with the heading: FRAME NUMBERS using the Home/Fill commands.

To organize your data effectively, start by creating a new column titled "Duplicate Frame Numbers" next to your original frame numbers Next, in a separate column to the right of the duplicate frame numbers, utilize the =RAND() function to generate random numbers corresponding to each frame number Ensure that the format of this column is adjusted to display each random number with three decimal places for clarity.

(d) Sort the duplicate frame numbers and random numbers into a random order (e) Print the result so that the spreadsheet fits onto one page

(f) Circle on your printout the I.D number of the first 15 children that you would call in your phone survey

(g) Save the file as: RAND9

It's important to note that each time the RAND() function is used in Excel, it generates a unique random order of children ID numbers Consequently, the sequence of random numbers provided in this Excel Guide will differ from the one you create, which is completely normal and expected.

To conduct a random sample of registered voters for a political poll in Missouri, you would select 10 out of the state's 114 counties This information is based on data from the U.S Census Bureau, which confirms that Missouri has 114 counties, while the entire United States comprises 3,140 counties across all 50 states.

(a) Set up a spreadsheet of frame numbers for these counties with the heading: FRAME NO.

To organize your data effectively, first create a column labeled "Duplicate frame no." adjacent to your original frame numbers Next, add another column titled "Random number" to the right of the duplicate frame numbers, utilizing the =RAND() function to generate random values for each frame number Finally, adjust the formatting of the random number column to display three decimal places for each entry.

(f) Circle on your printout the I.D number of the first ten counties that the pollster would call in his phone survey

(g) Save the file as: RANDOM6

The Sales department at the company aims to conduct a customer satisfaction survey targeting 20 of its 76 key accounts A key account is defined by the Sales Vice-President as any customer who has made purchases totaling at least $30,000 within the last 90 days.

(a) Set up a spreadsheet of frame numbers for these customers with the heading: FRAME NUMBERS.

To organize your data, first create a column labeled "Duplicate Frame Numbers" adjacent to your original frame numbers Next, add another column titled "Random Number" to the right of the duplicate frame numbers In this column, utilize the =RAND() function to generate random numbers corresponding to each entry in the duplicate frame numbers column Finally, adjust the formatting of this column to display each random number with three decimal places.

(f) Circle on your printout the I.D number of the first 20 customers that the Sales Vice-President would call for his phone survey.

(g) Save the file as: RAND5

Frankfort-Nachmias, C and Nachmias, D Research Methods in the Social Sciences (7 th ed.). New York, NY: Worth Publishers, 2008.

In their 2000 paper presented at the American Political Science Association annual meeting, Howell, Wolf, Campbell, and Peterson examined the impact of school vouchers on test scores in Dayton, Ohio, New York City, and Washington, D.C Their research utilized randomized field trials to provide evidence on the effectiveness of school voucher programs in improving educational outcomes.

U.S Census Bureau Census 2000 PHC-T-4 Ranking tables for counties 1990 and 2000 Retrieved from http://www.census.gov/population/www/cen2000/briefs/phc-t4/tables/tab01.pdf

Confidence Interval About the Mean Using the TINV Function and Hypothesis Testing

This chapter focuses on two ideas: (1) finding the 95 % confidence interval about the mean, and (2) hypothesis testing.

Let’s talk about the confidence interval first.

Confidence Interval About the Mean

How to Estimate the Population Mean

Objective: To estimate the population mean,μ

The population mean represents the average value among individuals in a specific target group For instance, if we wanted to assess the preference of adults aged 25–44 for a new Ben & Jerry’s ice cream flavor, surveying every individual in that demographic across the U.S would be impractical due to the extensive time and high costs involved in such a study.

Instead of testing the entire population, we can take a sample to estimate the population mean, which is a more efficient use of time and resources This approach, known as "inferential statistics," allows us to infer the population mean based on the sample mean (King et al., 1994).

When we study a sample of people in social science research, we know the size of our sample (n), the mean of our sample X

, and the standard deviation of our sample (STDEV) We use these figures to estimate the population mean with a test called the “confidence interval about the mean.”

Estimating the Lower Limit and the Upper Limit

of the 95 % Confidence Interval About the Mean

The theoretical background of this test is beyond the scope of this book, and you can learn more about this test from studying any good statistics textbook (e.g Levine

2011and Pollock2009) but the basic ideas are as follows.

We assume that the population mean is somewhere in an interval which has a

In this book, we define a "lower limit" and an "upper limit" for our interval, aiming for a 95% confidence level that the population mean falls within this range.

“We are 95 % confident that the population mean in miles per gallon (mpg) for the Chevy Impala automobile is between 26.92 miles per gallon and 29.42 miles per gallon.”

We can confidently advertise this car as achieving 28 miles per gallon (mpg) since this figure falls within the 95% confidence interval derived from our research, which ranges from 26.92 mpg to 29.42 mpg Although the exact population mean is unknown, we are assured that it lies within this interval, validating our claim.

But we are only 95 % confident that the population mean is inside this interval, and 5 % of the time we will be wrong in assuming that the population mean is

In social science research, we typically aim for a 95% confidence level in our assumptions, which is a standard yet arbitrary choice While we could opt for different confidence levels such as 80%, 90%, or even 99%, this book will consistently maintain a 95% confidence threshold This approach eliminates any uncertainty regarding the level of confidence you should have when tackling the problems presented in this book.

So how do we find the 95 % confidence interval about the mean for our data?

In words, we will find this interval this way:

To calculate the confidence interval, first determine the sample mean and the standard error of the mean (s.e.) Next, add 1.96 times the standard error to the sample mean to establish the upper limit of the confidence interval Conversely, subtract 1.96 times the standard error from the sample mean to find the lower limit of the confidence interval.

40 3 Confidence Interval About the Mean Using the TINV

The standard error of the mean (s.e.) is calculated by dividing the sample's standard deviation (STDEV) by the square root of the sample size (n).

In mathematical terms, the formula for the 95 % confidence interval about the mean is:

To calculate the confidence interval, first determine the upper limit by adding 1.96 times the standard error (s.e.) to the mean, and then find the lower limit by subtracting 1.96 times the s.e from the mean The term 1.96 s.e represents the product of 1.96 and the standard error of the mean, which is essential for constructing the confidence interval.

Note:We will explain shortly where the number 1.96 came from.

Let’s try a simple example to illustrate this formula.

3.1.3 Estimating the Confidence Interval for the Chevy

Impala in Miles Per Gallon

If Chevy Impala owners meticulously record their mileage along with the gallons consumed for two full tanks of gas, they can effectively assess their vehicle's fuel efficiency and performance This tracking not only provides valuable insights into their driving habits but also helps in identifying potential areas for improvement in fuel consumption By analyzing this data, Impala owners can make informed decisions to enhance their vehicle's overall efficiency and save on fuel costs.

A study involving 49 vehicle owners revealed an average fuel efficiency of 27.83 miles per gallon (mpg), with a standard deviation of 3.01 mpg The calculated standard error (s.e.) for this sample is 0.43, derived from dividing the standard deviation by the square root of the sample size.

The 95 % confidence interval for these data would be:

Theupper limit of this confidence intervaluses the plus sign of thesign in the formula Therefore, the upper limit would be:

Similarly, the lower limit of this confidence interval uses the minus sign of thesign in the formula Therefore, the lower limit would be:

The result of our research study would, therefore, be the following:

“We are 95 % confident that the population mean for the Chevy Impala is somewhere between 26.99 mpg and 28.67 mpg.”

3.1 Confidence Interval About the Mean 41

Based on our data, we can confidently state that this car achieves 28 miles per gallon (mpg), as this figure falls within the 95% confidence interval for the population mean.

You are probably asking yourself: “Where did that 1.96 in the formula come from?”

3.1.4 Where Did the Number “1.96” Come From?

A detailed mathematical answer to that question is beyond the scope of this book, but here is the basic idea.

The assumption of a "normally distributed" population implies that if all individuals were tested, the data would form a "normal curve," resembling the outline of the Liberty Bell in Philadelphia This curve is symmetric, meaning that if it were divided down the middle and folded over, the two halves would align perfectly For further insights into the normal curve, refer to Steinberg (2008) and Frankfort-Nachmias and Nachmias (2008).

Integral calculus is not covered in this book, but we aim to determine the lower and upper limits of population data within the normal curve, ensuring that 95% of the area falls between these limits For research studies involving more than 40 participants, these limits are calculated as plus or minus 1.96 times the standard error of the mean (s.e.) of the sample This calculation provides the confidence interval's upper and lower bounds For further understanding, refer to a reputable statistics textbook, such as Salkind (2010).

The number 1.96 would change if we wanted to be confident of our results at a different level from 95 % as long as we have more than 40 people in our research study.

1 If we wanted to be 80 % confident of our results, this number would be 1.282.

In this book, we aim for 95% confidence in our results, which is why we will consistently use a value of 1.96 for research studies involving more than 40 participants.

The value of 1.96 in the confidence interval for the mean is not a constant; it varies depending on the desired confidence level.

3.1.5 Finding the Value for t in the Confidence

Objective: To find the value for t in the confidence interval formula

The correct formula for the confidence interval about the mean for different sample sizes is the following:

To calculate the 95% confidence interval, start by determining the sample mean (X) The upper limit is found by adding the product of the t-value and the standard error (s.e.) to the sample mean, while the lower limit is obtained by subtracting the same product from the sample mean The t-value can be located in the table provided in Appendix E of this book.

Objective: To find the value of t in the t-table in AppendixE

Before we get into an explanation of what is meant by “the value of t,” let’s give you practice in finding the value of t by using the t-table in AppendixE.

Keep your finger on Appendix Eas we explain how you need to “read” that table.

In this chapter, the test referred to as the "confidence interval about the mean test" requires you to consult the first column on the left in Appendix E to determine the critical value of t for your research study, which is labeled as "sample size n."

Where Did the Number “1.96” Come From?

A detailed mathematical answer to that question is beyond the scope of this book, but here is the basic idea.

We assume that population data is normally distributed, resembling a "normal curve" if all individuals were tested This curve, which is symmetric like the outline of the Liberty Bell in Philadelphia, can be perfectly folded in half, with both sides matching For further insights into the normal curve, refer to Steinberg (2008) and Frankfort-Nachmias and Nachmias (2008).

This article focuses on determining the confidence interval for population data within a normal curve, specifically aiming to capture 95% of the area under the curve For research studies involving more than 40 participants, the confidence interval limits are calculated as plus or minus 1.96 times the standard error of the mean (s.e.) This calculation provides the upper and lower limits of the confidence interval, which is essential for accurate data interpretation For further exploration of this concept, readers are encouraged to refer to reputable statistics literature, such as Salkind (2010).

The number 1.96 would change if we wanted to be confident of our results at a different level from 95 % as long as we have more than 40 people in our research study.

In this book, we aim for 95% confidence in our results; therefore, whenever our research study involves more than 40 participants, we will consistently use the value of 1.96.

You might be wondering if the confidence interval for the mean is always 1.96 The answer is no, and we will clarify the reasons behind this.

Finding the Value for t in the Confidence

Objective: To find the value for t in the confidence interval formula

The correct formula for the confidence interval about the mean for different sample sizes is the following:

To calculate the 95% confidence interval, start by determining the sample mean (X) The upper limit is found by adding the product of the t-value and the standard error (s.e.) to the sample mean, while the lower limit is obtained by subtracting this product from the sample mean You can find the appropriate t-value in the table located in Appendix E of this book.

Objective: To find the value of t in the t-table in AppendixE

Before we get into an explanation of what is meant by “the value of t,” let’s give you practice in finding the value of t by using the t-table in AppendixE.

Keep your finger on Appendix Eas we explain how you need to “read” that table.

In this chapter, you will utilize the “confidence interval about the mean test” to determine the critical value of t for your research study by referring to the first column labeled “sample size n” in Appendix E.

To determine the value of t for your research study, locate the sample size in the first column of the table, then move to the right to find the corresponding t value in the "critical t column," which is used for a 95% confidence interval about the mean For instance, if your study includes 14 participants, the t value will be 2.160.

If you have 26 people in your research study, the value of t is 2.060.

In research studies with over 40 participants, the t value is consistently 1.96, which is essential for achieving 95% confidence in your statistical results The "critical t column" found in Appendix E provides the necessary t value to determine the significance of your findings This book assumes that you aim for a 95% confidence level in your statistical tests, making the t values in Appendix E crucial for calculating the 95% confidence interval around the mean.

To calculate the confidence interval for the mean using Excel, you first need to determine the value of t Once you have this value, you can apply it within Excel's functions to compute the confidence interval effectively This process allows you to analyze your data and derive meaningful insights with precision.

Using Excel ’ s TINV Function to Find the Confidence

Objective: To use the TINV function in Excel to find the confidence interval about the mean

When you use Excel, the formulas for finding the confidence interval are:

Lower limit: ẳXTINVð10:95,n1ị*s:e: ðno spaces between these symbolsị ð3:3ị

Upper limit: ẳXỵTINVð10:95,n1ị*s:e: ðno spaces between these symbolsị ð3:4ị

In Excel formulas, the asterisk symbol (*) indicates multiplication, representing the term "times." Additionally, as mentioned in Chapter 1, 'n' refers to the sample size, while 's' denotes the sample size minus one.

In Chapter 1, we learned that the standard error of the mean (s.e.) is calculated by dividing the standard deviation (STDEV) by the square root of the sample size (n) To illustrate this concept, we will use Excel to calculate the 95% confidence interval for the mean in a practical example.

General Motors claims that the Chevy Impala has a highway fuel efficiency of 28 miles per gallon (mpg), which we will refer to as the "reference value" for this vehicle.

As an employee at Ford Motor Co., you aim to verify a specific claim through research evidence To achieve this, you collect relevant data and apply a two-sided 95% confidence interval to assess the mean, ensuring that your findings are statistically reliable.

Using Excel to Find the 95 % Confidence Interval

Objective: To analyze the data using a two-side 95 % confidence interval about the mean

In a study involving 25 new car owners, participants tracked their mileage over two tanks of gas and recorded the average miles per gallon achieved The findings are illustrated in Fig 3.1, showcasing the performance metrics of the vehicles under review.

To analyze the provided data, create a spreadsheet in Excel and calculate the sample size (n), mean, standard deviation (STDEV), and standard error of the mean (s.e.) using the specified cell references.

Enter the other mpg data in cells A7:A30

Now, highlight cells A6:A30 and format these numbers in number format (one decimal place) Center these numbers in Column A Then, widen columns A and

B by making both of them twice as wide as the original width of column

A Then, widen column C so that it is three times as wide as the original width of column A so that your table looks more professional.

Fig 3.1 Worksheet Data for Chevy Impala (Practical Example)

B26: Draw a picture below this confidence interval

B28: ‘26.92 - (be sure to add the single quotation mark before 26.92 so that Excel treats this as a label, instead of a number)

To properly label cells in Excel, begin with a single quotation mark before the text For instance, in cell C28, input ‘ - 28 -–28.17, in D28 use ‘ -, and in E28 enter ‘29.42 This ensures that Excel interprets these entries as labels rather than numbers.

Fig 3.2 Example of Chevy Impala Format for the Confidence Interval About the Mean Labels

Now, align the labels underneath the picture of the confidence interval so that they look like Fig.3.3.

Next, name the range of data from A6:A30 as: miles

D7: Use Excel to find the sample size

D10: Use Excel to find the mean

D13: Use Excel to find the STDEV

D16: Use Excel to find the s.e.

Now, you need to find the lower limit and the upper limit of the 95 % confidence interval for this study.

We will use Excel’s TINV function to do this We will assume that you want to be 95 % confident of your results.

F21: ẳD10TINV 1ð :95, 24ị*D16 Fig 3.3 Example of Drawing a Picture of a Confidence Interval About the Mean Result

Note that this TINV formula uses 24 since 24 is one less than the sample size of

25 (i.e., 24 is n1) Note that D10 is the mean, while D16 is the standard error of the mean The above formula gives thelower limit of the confidence interval, 26.92.

The upper limit of the confidence interval is 29.42, while the lower limit is 26.92 To ensure clarity in your Excel spreadsheet, apply number formatting with two decimal places for the mean, standard deviation, standard error of the mean, and both limits of the confidence interval Be mindful that if you print the spreadsheet now, the lower and upper limits may extend onto a second page due to their size, requiring adjustments for proper presentation.

To optimize your spreadsheet for printing, utilize Excel's "Scale to Fit" feature found in the Page Layout tab to reduce the size to 95% of its current dimensions After applying this adjustment, you'll observe that the dotted line next to the measurements of 26.92 and 29.42 signifies that these dimensions will now fit onto a single printed page, as illustrated in Figure 3.4.

Fig 3.4 Result of Using the TINV Function to Find the Confidence Interval About the Mean

Note that you have drawn a picture of the 95 % confidence interval beneath cell B26, including the lower limit, the upper limit, the mean, and the reference value of

28 mpg given in the claim that the company wants to make about the car’s miles per gallon performance.

Now, let’s write the conclusion to your research study on your spreadsheet:

C33: Since the reference value of 28 is inside

C34: the confidence interval, we accept that

C35: the Chevy Impala does get 28 mpg.

Your research study accepted the claim that the Chevy Impala did get 28 miles per gallon on the highway The average miles per gallon in your study was 28.17 (See Fig.3.5).

Save your resulting spreadsheet as: CHEVY7

Fig 3.5 Final Spreadsheet for the Chevy Impala Confidence Interval About the Mean

Hypothesis Testing

Hypotheses Always Refer to the Population of People

or Events That You Are Studying

The first step is to understand that our hypotheses always refer to thepopulationof people under study.

When studying 18–24 year-olds in St Louis, it is crucial to select a representative sample from this demographic The goal is to ensure that the findings of the study can be generalized to all individuals aged 18–24 in St Louis, rather than being limited to just those in the chosen sample.

Our study focuses on the population of 18–24 year-olds in St Louis, with a specific subset known as the sample representing this demographic.

Our sample sizes usually consist of only a small number of individuals, so we focus on the results primarily for their potential to be generalized to the broader population of interest.

That is why our hypotheses always refer to the population, and never to the sample of people in our study.

You will recall from Chap.1that we used the symbol:Xto refer to the mean of the sample we use in our research study (see Sect.1.1).

We will use the symbol:μ(the Greek letter “mu”) to refer to thepopulation mean.

In testing our hypotheses, we are trying to decide which one of two competing hypothesesabout the population meanwe should accept given our data set.

The Null Hypothesis and the Research (Alternative)

The two main hypotheses in statistical analysis are the null hypothesis (H0) and the research hypothesis (H1), which is also known as the alternative hypothesis.

Let’s explain first what is meant by the null hypothesis and the research hypothesis:

1 The null hypothesis is what we accept as true unless we have compelling evidence that it is not true.

2 The research hypothesis is what we accept as true whenever we reject the null hypothesis as true.

In the American legal system, there is a fundamental principle that individuals are presumed innocent until proven guilty by a jury This concept establishes the null hypothesis, which posits that the defendant is innocent, in contrast to the research hypothesis that asserts the defendant's guilt.

In Missouri, the state slogan "Show me" reflects the residents' skepticism and demand for evidence, emphasizing their belief that actions are more significant than words This attitude signifies that Missourians prefer to verify claims through tangible proof rather than mere assertions, highlighting a strong cultural value placed on accountability and authenticity.

Hypothesis testing involves evaluating two competing statements: the null hypothesis and the research hypothesis Since both cannot be true simultaneously, the goal is to determine which hypothesis to accept as valid and which to reject based on statistical analysis.

In social science research, rating scales are commonly employed to assess individuals' attitudes towards a company, its products, or their purchase intentions These scales typically consist of 5-point, 7-point, or 10-point formats, although variations in scale values may also be utilized.

3.2.2.1 Determining the Null Hypothesis and the Research Hypothesis

When Rating Scales Are Used

Here is a typical example of a 7-point scale in education for parents of 8 th grade pupils at the end of a school year (see Fig.3.6):

So, how do we decide what to use as the null hypothesis and the research hypothesis whenever rating scales are used?

Objective: To decide on the null hypothesis and the research hypothesis whenever rating scales are used.

In order to make this determination, we will use a simple rule:

Rule:Whenever rating scales are used, we will use the “middle” of the scale as the null hypothesis and the research hypothesis.

In the above example, since 4 is the number in the middle of the scale (i.e., three numbers are below it, and three numbers are above it), our hypotheses become:

If our statistical test reveals that the population mean for the attitude scale item is approximately 4, we conclude that we accept the null hypothesis This suggests that parents of 8th-grade students are neither satisfied nor dissatisfied with the quality of the academic programs provided by their child's school.

If our statistical test shows that the population mean significantly differs from 4, we reject the null hypothesis and accept the research hypothesis.

Parents of 8th grade students expressed high levels of satisfaction with the academic programs provided by their children's school, as indicated by a sample mean that significantly exceeds the anticipated population mean of 4.

Fig 3.6 Example of a Rating Scale Item for Parents of 8th Graders (Practical Example)

52 3 Confidence Interval About the Mean Using the TINV or

Parents of eighth-grade students expressed considerable dissatisfaction with the academic programs provided by their children's school, particularly when the sample mean of their feedback was notably lower than the anticipated population mean of 4.

Both of these conclusions cannot be true We accept one of the hypotheses as

“true” based on the data set in our research study, and the other one as “not true” based on our data set.

A social science researcher must determine whether to accept the null hypothesis or the research hypothesis as true based on the data collected in their study.

Let’s try some examples of rating scales so that you can practice figuring out what the null hypothesis and the research hypothesis are for each rating scale.

In the spaces in Fig.3.7, write in the null hypothesis and the research hypothesis for the rating scales:

Fig 3.7 Examples of Rating Scales for Determining the Null Hypothesis and the Research Hypothesis

Here are the answers to these three questions:

1 The null hypothesis isμẳ3, and the research hypothesis isμ6ẳ3 on this 5-point scale (i.e the “middle” of the scale is 3).

2 The null hypothesis isμẳ4, and the research hypothesis isμ6ẳ4 on this 7-point scale (i.e., the “middle” of the scale is 4).

3 The null hypothesis isμẳ5.5, and the research hypothesis isμ6ẳ5.5 on this 10-point scale (i.e., the “middle” of the scale is 5.5 since there are 5 numbers below 5.5 and 5 numbers above 5.5).

As another example, Holiday Inn Express in its Stay Smart Experience Survey uses 4-point scales where:

On this scale, the null hypothesis is: μẳ2.5 and the research hypothesis is: μ6ẳ2.5, because there are two numbers below 2.5, and two numbers above 2.5 on that rating scale.

Now, let’s discuss the 7 STEPS of hypothesis testing for using the confidence interval about the mean.

The 7 Steps for Hypothesis-Testing Using the Confidence

Objective: To learn the 7 steps of hypothesis-testing using the confidence interval about the mean

There are seven basic steps of hypothesis-testing for this statistical test.

3.2.3.1 STEP 1: State the Null Hypothesis and the Research Hypothesis

When utilizing numerical scales in surveys, it's crucial to focus on the midpoint of the scale For instance, in a 7-point scale ranging from 1 (poor) to 7 (excellent), the hypotheses should center around the middle values of the scale.

3.2.3.2 STEP 2: Select the Appropriate Statistical Test

In this chapter we are studying the confidence interval about the mean, and so we will select that test.

3.2.3.3 STEP 3: Calculate the Formula for the Statistical Test

You will recall (see Sect.3.1.5) that the formula for the confidence interval about the mean is:

In this chapter, we outlined the procedure for calculating the confidence interval for the mean using Excel The steps involved in applying this formula are essential for accurate data analysis.

1 Use Excel’sẳCOUNT function to find the sample size.

2 Use Excel’sẳAVERAGE function to find the sample mean,X.

3 Use Excel’sẳSTDEV function to find the standard deviation, STDEV.

4 Find the standard error of the mean (s.e.) by dividing the standard deviation (STDEV) by the square root of the sample size, n.

5 Use Excel’s TINV function to find the lower limit of the confidence interval.

6 Use Excel’s TINV function to find the upper limit of the confidence interval.

3.2.3.4 STEP 4: Draw a Picture of the Confidence Interval About the Mean, Including the Mean, the Lower Limit of the Interval, the Upper Limit of the Interval, and the Reference Value Given in the Null Hypothesis, H 0

3.2.3.5 STEP 5: Decide on a Decision Rule

(a) If the reference value is inside the confidence interval, accept the null hypothesis, H0

(b) If the reference value is outside the confidence interval, reject the null hypothesis, H0, and accept the research hypothesis, H1

3.2.3.6 STEP 6: State the Result of Your Statistical Test

When using a confidence interval to estimate the mean, there are two potential outcomes, but only one can be deemed "true." Thus, your results will fall into one of these two categories.

Either:Since the reference value is inside the confidence interval, we accept the null hypothesis, H0

Or: Since the reference value is outside the confidence interval, we reject the null hypothesis, H0, and accept the research hypothesis, H1

3.2.3.7 STEP 7: State the Conclusion of Your Statistical

Summarizing the results of a statistical test in simple, concise language can be challenging, especially when aiming to communicate effectively with individuals unfamiliar with statistics This article focuses on how to articulate the conclusion of a confidence interval regarding the mean test, ensuring clarity for all readers, including those without a statistics background Throughout this book, we will provide ample practice to master this crucial skill.

Let’s set some basic rules for sating the conclusion of a hypothesis test.

Rule #1: Whenever you reject H0and accept H1, you must use the word “significantly” in the conclusion to alert the reader that this test found an important result.

Rule #2: Create an outline in words of the “key terms” you want to include in your conclusion so that you do not forget to include some of them.

Rule #3: Write the conclusion in plain English so that the reader can understand it even if that reader has never taken a statistics course.

Let’s practice these rules using the Chevy Impala Excel spreadsheet that you created earlier in this chapter, but first we need to state the hypotheses for that car.

If General Motors wants to claim that the Chevy Impala gets 28 miles per gallon on a billboard ad, the hypotheses would be:

Based on the analysis, the reference value of 28 mpg falls within the 95% confidence interval for the data collected, leading us to accept the null hypothesis (H0) for the Chevy Impala, confirming that the vehicle achieves an average fuel efficiency of 28 mpg.

Objective: To state the result when you accept H 0

Result:Since the reference value of 28 mpg is inside the confidence interval, we accept the null hypothesis, H0

Let’s try our three rules now:

Objective: To write the conclusion when you accept H 0

In this chapter, we adhere to a fundamental rule: if the reference value falls within the confidence interval, we must refrain from using the term "significantly" in our conclusions This guideline is consistently applied across all problems discussed.

Rule #2: The key terms in the conclusion would be:

Rule #3: The Chevy Impala did get 28 mpg.

Writing a conclusion after accepting the null hypothesis (H0) is simple, as it reflects the initial statements made in the null hypothesis Conversely, formulating a conclusion upon rejecting H0 and accepting the alternative hypothesis (H1) is more complex To enhance understanding, we will practice crafting such conclusions through three illustrative case examples.

Objective: To write the result and conclusion when you reject H 0

CASE #1: Suppose that an ad inThe Wall Street Journalclaimed that the Honda

Accord Sedan gets 34 miles per gallon The hypotheses would be:

Suppose that your research yields the following confidence interval:

30 31 32 34 lower Mean upper Ref. limit limit Value

Result: Since the reference value is outside the confidence interval, we reject the null hypothesis and accept the research hypothesis

The three rules for stating the conclusion would be:

Rule #1: We must include the word “significantly” since the reference value of

34 is outside the confidence interval.

Rule #2: The key terms would be:

– either “more than” or “less than”

Rule #3: The Honda Accord Sedan got significantly less than 34 mpg, and it was probably closer to 31 mpg.

The conclusion indicates that the miles per gallon (mpg) was below 34, as the sample mean was recorded at only 31 mpg Additionally, it is important to clarify that simply stating “significantly less than” after rejecting the null hypothesis does not provide a complete understanding of the results.

34 mpg,”because that does not tell the reader “how much less than 34 mpg” the sample mean was from 34 mpg To make the conclusion clear, you need to add:

“probably closer to 31 mpg” since the sample mean was only 31 mpg.

CASE #2: Suppose that you have been hired as a consultant by the St Louis

Symphony Orchestra (SLSO) to analyze the data from an Internet survey of attendees for a concert in Powell Symphony Hall in

St Louis last month You have decided to practice your data analysis skills on Question #7 given in Fig.3.8:

The hypotheses for this one item would be:

The null hypothesis posits that a mean score of 4 indicates attendees were neither satisfied nor dissatisfied with their SLSO concerts If the analysis reveals that the obtained mean score falls within a confidence interval that does not significantly differ from 4, it suggests a neutral overall sentiment among concertgoers.

1.8 _2.8 _3.8 4 lower Mean upper Ref. limit limit Value

Fig 3.8 Example of a Survey Item Used by the St Louis Symphony Orchestra (SLSO)

Result:Since the reference value is outside the confidence interval, we reject the null hypothesis and accept the research hypothesis

Rule #1: You must include the word “significantly” since the reference value is outside the confidence interval

– either satisfied or dissatisfied (since the result is significant)

Rule #3: Attendees were significantly dissatisfied, overall, on last month’s Internet survey with their experiences at concerts of the SLSO.

Note that you need to use the word “dissatisfied” since the sample mean of 2.8 was on the dissatisfied side of the middle of the rating scale.

A U.S Senator from Missouri has requested his staff to carry out a phone survey targeting registered voters in the state to gauge their opinions on his performance The staff conducts the survey using a random sample of voters, employing the specified survey item to gather insights on the Senator's effectiveness in office.

This item would have the following hypotheses:

Suppose that your research produced the following confidence interval for this item on the survey:

Fig 3.9 Example of a Survey Item for a U.S Senator from Missouri

Result:Since the reference value is outside the confidence interval, we reject the null hypothesis and accept the research hypothesis

The three rules for stating the conclusion would be:

Rule #1: You must include the word “significantly” since the reference value is outside the confidence interval

Rule #3: Registered voters in Missouri significantly approved of Senator ’s performance in the U.S Senate in a phone survey.

Since the mean rating of the Senator’s performance (5.8) was significantly greater than 5.5 on the approve side of the scale, we would say “significantly approved” to indicate this fact.

This chapter includes three practice problems designed to enhance your skills in articulating the conclusions of your results Additionally, this book provides numerous examples to support you in crafting clear and precise conclusions for your research findings.

Alternative Ways to Summarize the Result

The 7 STEPS for Hypothesis-Testing Using

The 9 STEPS for Hypothesis-Testing Using

Formula #1: Both Groups Have More than 30 People

What Is a “Correlation?”

Creating a Chart and Drawing the Regression Line

Finding the Regression Equation

Printing Only Part of a Spreadsheet Instead

Testing the Difference Between Two Groups Using

Tiêu đề	Excel 2013 for Social Sciences Statistics A Guide to Solving Practical Problems
Tác giả	Thomas J. Quirk
Trường học	Springer International Publishing
Chuyên ngành	Social Sciences Statistics
Thể loại	guide
Năm xuất bản	2015
Thành phố	Switzerland

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Số trang	267
Dung lượng	12,24 MB