Excel 2016 for engineering statistics a guide to solving practical problems

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 upset about not understanding averages, I realized the importance of explaining this concept clearly.

Jennifer asked for help with her math, and I explained that to find the average, you need to sum all the scores and divide by the total count of numbers However, she didn't appreciate my casual tone and responded with a serious look, insisting, "Dad, this is serious!" indicating that she believed I was joking rather than offering genuine assistance.

“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 easily arrive at the correct answer, as Excel will automate 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.”

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

XẳΣX n ð1:1ị © Springer International Publishing Switzerland 2016

T.J Quirk, Excel 2016 for Engineering Statistics, Excel for Statistics,

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

Suppose that you had these six engineering test scores on an 7-item true-false quiz:

To find the mean of these scores, you add them up, and then divide by the number of scores So, the mean is: 25/6ẳ4.17

Standard Deviation

Standard deviation measures the proximity of scores to the mean, indicating how closely they cluster A small standard deviation signifies that 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 score (XX).

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

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

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 standard deviation using Excel rather than a calculator, as detailed in basic statistics resources like Schuenemeyer and Drew (2011) By applying Excel to a specific set of six numbers, we determine that the standard deviation, denoted as S, is 1.47.

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

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 data points in your dataset For instance, in the previous example, the standard error is 0.60, which can be verified using a calculator.

To understand the concepts of standard deviation and standard error of the mean, refer to McKillup and Dyar (2010) and Schuenemeyer and Drew (2011) This article will guide you on how to utilize Excel to calculate sample size, mean, standard deviation, and standard error of the mean, specifically analyzing the levels of sulfur dioxide in rainfall, measured in milligrams (mg) per liter (L) For context, one milligram is one-thousandth of a gram, and one liter is the volume equivalent to one kilogram of pure water under standard conditions We will consider data collected from eight rainfall samples, as illustrated in Fig 1.1.

Fig 1.1 Worksheet Data for Sulphur Dioxide Levels

1.3 Standard Error of the Mean 3

Sample Size, Mean, Standard Deviation, and Standard Error

Using the Fill/Series/Columns Commands

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

Home (top left of screen)

Important note: The “Paste” command should be on the top of your screen on the far left of the screen.

Important note: Notice the Excel commands at the top of your computer screen:

File!Home!Insert!Page Layout!Formulas!etc.

If these commands ever“disappear”when you are using Excel,you need to click on“Home”at the top left of your screen to make them reappear!

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

Fig 1.2 Home/Fill/Series commands

The sample numbers should be identified as 1–8, with 8 in cell B11.

Enter the milligrams per liter values in cells C4 to C11, ensuring to double-check your figures for accuracy to obtain the correct results.

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

Fig 1.3 Example of Dialogue Box for Fill/Series/Columns/Step Value/Stop Value commands1.4 Sample Size, Mean, Standard Deviation, and Standard Error of the Mean 5

To ensure all the information fits properly, you need to widen Column C on your computer screen.

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

Place your mouse pointer on your computer at the far right corner of C 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”

Take your finger off your mouse to set the new column width (see Fig.1.4)

Then, click on any empty cell (i.e., any blank cell) to “deselect” column C 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 B andColumn C.

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 pointer on B3 and drag it to the right and down to highlight cells B3:C11 so that these cells appear in a darker color

Fig 1.4 Example of How to Widen the Column Width

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

Click on this icon to center the information in the selected cells (see Fig.1.6)

To simplify referencing milligrams per liter in your formulas, it's beneficial to assign a name to your data range instead of recalling specific cell locations like C4:C11 For instance, you can label this group of cells as "Weight," or choose any name that suits your preference.

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

Naming a Range of Cells

Objective: To name the range of data for the milligrams per liter with the name:

Highlight cells C4: C11 by left-clicking your mouse pointer on C4 and dragging it down to C11

Formulas (top left of your screen)

Define Name (top center of your screen)

Weight (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 C4:C11

Now, add the following terms to your spreadsheet:

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

When using a formula in Excel, it is essential to start the formula with an equal sign (=) to indicate to Excel that you are entering a formula.

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 samples of rainfall in your sample.

Finding the Mean Score Using

Objective: To find the mean weight figure using theẳAVERAGE function

This command should insert the number 0.8125 into cell F9.

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

1.4 Sample Size, Mean, Standard Deviation, and Standard Error of the Mean 9

Finding the Standard Deviation Using

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

This command should insert the number 0.352288 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 0.124553 into cell F15 (see Fig.1.9).

Important reminder: Always verify that all figures in your spreadsheet are placed in the correct cells, as any discrepancies will cause the formulas to malfunction.

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

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 decimal places 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 00.0 until the option "Decrease Decimal" appears.

Click on this icontwiceand 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 Places1.4 Sample Size, Mean, Standard Deviation, and Standard Error of the Mean 11

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

Saving a Spreadsheet

Objective: To save this spreadsheet with the name: sulphur3

To ensure you can access your spreadsheet later, the first step is to determine the appropriate location for saving it You have multiple options for storage; if you choose to save it on your personal computer, you can store it on your hard drive If you're unsure how to do this, consider asking someone for assistance.

Or, you can save it onto a “CD” or onto a “flash drive.” You then need to complete these steps:

To save a file, simply scroll through the left sidebar to select your desired location, then click on the specific folder where you wish to save it, such as "This PC" or "My Documents."

File name: sulphur3 (enter this name to the right of File name; see Fig.1.12) Fig 1.11 Example of Converting Numbers to Two Decimal Places

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 1.12 Dialogue Box of Saving an Excel Workbook File as “sulphur3” in My Documents location

Print (top of your screen)

The final spreadsheet is given in Fig1.14

Fig 1.13 Example of How to Print an Excel Worksheet

Using the File/Print/Print

Before concluding this chapter, let’s practice adjusting figure formats in a spreadsheet through two examples: first, applying two decimal places for dollar amounts, and second, using three decimal places for numerical figures.

To close your spreadsheet, navigate to File and select Save, then choose Close To open a new blank Excel spreadsheet, go to File and click on New, followed by selecting Blank Workbook from the options on the far 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

Fig 1.14 Final Result of Printing an Excel Spreadsheet

1.7 Formatting Numbers in Currency Format (Two decimal places) 15

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

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

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

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

1.8 Formatting Numbers in Number Format (Three decimal places) 17

Ensure 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 select "Don't Save" since 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

Limonite, a mineral composed of various other minerals, significantly influences soil color and the weathered surfaces of rocks, and is also a source of iron ore To analyze the iron content in limonite samples, one could calculate the mean, standard deviation, and standard error of the mean based on hypothetical data, as illustrated in Fig 1.17.

To analyze the data, utilize Excel to calculate the sample size, mean, standard deviation, and standard error of the mean, ensuring to label each result clearly Round the mean, standard deviation, and standard error of the mean to two decimal places and apply number formatting for these values.

(b) Print the result on a separate page.

(c) Save the file as: iron3

Fig 1.17 Worksheet Data for Chap 1: Practice

As a research assistant, you are tasked with calculating the average lead concentration in air samples collected near Route 101 in San Francisco during weekday afternoons, specifically between 4 p.m and 7 p.m The analysis will focus on determining the concentration measured in micrograms per cubic meter (μg/m³), utilizing the hypothetical data provided in Fig 1.18.

To analyze the data effectively, create a table in Excel to organize the information Next, utilize Excel functions to calculate the sample size, mean, standard deviation, and standard error of the mean, ensuring to label each result clearly Remember to round the mean, standard deviation, and standard error to two decimal places using the appropriate number format for clarity.

(c) Save the file as: air3

In a study to assess the silver content in ore samples from a mine, 16 samples were collected from various locations within the mine Each sample was processed to quantify the amount of silver present, with the results illustrated in Fig.1.19.

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

1.9 End-of-Chapter Practice Problems 19

To analyze the given data in Excel, first create a table to organize the information Next, calculate the sample size, mean, standard deviation, and standard error of the mean, placing these results to the right of the table Ensure that each value is clearly labeled, and round the mean, standard deviation, and standard error of the mean to three decimal places using the number format for clarity.

(c) Save the file as: SILVER3

McKillup S., Dyar M Geostatistics Explained: an introductory guide for earth scientists Cam- bridge: Cambridge University Press; 2010.

Schuenemeyer J, Drew L Statistics for Earth and Environmental Scientists Hoboken: John Wiley

Fig 1.19 Worksheet Data for Chap 1: Practice

Salt marshes are vital coastal wetlands located along the eastern seaboard of the USA, where freshwater meets seawater These unique ecosystems experience tidal flooding, requiring the resident plants to adapt to the saline conditions.

Salinity, or the salt content of water, is influenced by the proximity of a marsh to the ocean A biogeographer examining the impact of salinity on vegetation in a Maine salt marsh has conducted a detailed mapping of the area.

To conduct a study on salinity levels in a salt marsh, you need to randomly sample 5 out of 32 distinct geographic areas This process requires defining a "sampling frame" to ensure accurate representation Utilizing Excel skills, you can effectively select these areas for analysis, allowing for a comprehensive measurement of salinity percentages across the chosen sites.

A sampling frame is essential for selecting a random sample, and in this case, it consists of 32 distinct areas within a salt marsh Each area is assigned a unique identification code, starting with 1 for the first area, 2 for the second, and continuing sequentially up to 32 for the last area This structured approach ensures that every area is accounted for in the sampling process.

32 with each area having a unique ID number.

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

Creating Frame Numbers for Generating Random Numbers

Objective: To create the frame numbers for generating random numbers

To create frame numbers in column A, use the Home/Fill commands as detailed in Section 1.4.1 of this book Start by filling the cells with numbers ranging from 1 to 32, ensuring that the number 32 is placed in cell A35 Follow the outlined steps to achieve this efficiently.

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, apply 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 data, and then center the content within both columns This will yield a layout similar to the one shown in Fig 2.4.

Fig 2.3 Frame Numbers from 1 to 32

2.1 Creating Frame Numbers for Generating Random Numbers 23

Save this file as: Random30

You may be questioning the duplication of information in both Column A and Column B of your spreadsheet This approach ensures that you have precisely 32 frame numbers before you sort them into a random sequence, allowing for an accurate and organized final result.

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

Creating Random Numbers in an Excel Worksheet

(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 generate a random number in a cell, ensure you include both an open parenthesis and a closed parenthesis after the RAND() function The RAND command evaluates the cells to the left of its location and assigns a random number to the cell containing the RAND() command.

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, 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:

Random Numbers Assigned to the Duplicate Frame

2.2 Creating Random Numbers in an Excel Worksheet 25

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:

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!

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

2.3 Sorting Frame Numbers into a Random Sequence 27

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

5 of the 32 areas of the salt marsh, you now can do that by selecting thefirst five ID numbersin DUPLICATE FRAME NO column after the sort.

While your initial set of five random numbers may differ from the ones chosen in our random sorting process in this chapter, we will identify these five area IDs using Figure 2.9.

When using the RAND() function in Excel, it's important to note that the five ID numbers generated from your random selection will differ from those shown in Fig 2.9, as Excel produces a new set of random numbers each time the command is executed.

Before concluding this chapter, it's essential to understand how to print a file effectively, ensuring that all its content fits onto a single page without spilling over onto additional pages.

2.4 Printing an Excel File So That All of the Information Fits onto One Page

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

2.4 Printing an Excel File So That All of the Information Fits onto One Page 29

This chapter includes three practice problems that involve sorting random numbers, specifically for files containing 63 resistors, 114 steel samples, and 75 toxic waste sites To ensure these files can be printed on a single page, it is essential to format them appropriately, as they may be too large to fit 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 at its current format will result in the name "Jennifer" spilling over onto a second page due to its overflow beyond the designated page range.

To ensure that all information, including the name Jennifer, fits on a single printed page, 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)

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

In the displayed document, the name "Jennifer" appears on the second page, positioned below the horizontal dotted line, as illustrated in Fig 2.11 This dotted line indicates the outline dimensions of the file, which would be relevant if the document were printed at this moment.

Fig 2.10 Dialogue Box for Page Layout/Scale to Fit Commands

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

Save the file as: Random34

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

Fig 2.11 Example of Scale Reduced to 95 % with “Jennifer” to be Printed on a Second Page

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)

2.4 Printing an Excel File So That All of the Information Fits onto One Page 31

End-of-Chapter Practice Problems

To ensure quality control, an electronics company must randomly test 15 out of 63 electrical resistors of a specific type This sampling process is essential for evaluating the performance and reliability of the resistors in question.

Spreadsheet of 90 % Scale to Fit

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

To organize your spreadsheet effectively, start by creating a column titled "Duplicate Frame Numbers" next to your original frame numbers In the next column, utilize the =RAND() function to generate random numbers corresponding to each duplicate frame number, ensuring to format these numbers to display three decimal places Next, randomize the order of both the duplicate frame numbers and their associated random numbers Finally, adjust the layout so that the entire spreadsheet fits neatly onto a single page for printing.

(f) Circle on your printout the I.D number of the first 15 resistors that you would use in your research study

(g) Save the file as: RAND9

It's important to understand that each time the RAND() function is executed in Excel, it generates a unique random order of resistor 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.

As a consultant tasked with evaluating building materials for engineers designing suspension bridges, I have been provided with 114 samples of a new type of steel intended for future construction projects My objective is to conduct tensile strength tests on a random sample of 10 of these steel samples to assess their material consistency and performance suitability for bridge applications.

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

(b) Then, create a separate column to the right of these frame numbers which duplicates these frame numbers with the title: Duplicate frame no.

To generate random numbers for duplicate frame numbers, create a new column labeled "Random Number" next to the duplicate frame numbers column Utilize the =RAND() function to assign a random number to each frame number Finally, adjust the column format to display each random number with three decimal places.

(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 10 steel samples that would be used in this research study.

(g) Save the file as: RANDOM6

3 Suppose that an engineering field researcher wants to take a random sample of

20 of 75 toxic waste sites that have been mapped surrounding a commercial

A researcher is conducting a field study to assess the level of lead contamination in the soil surrounding a closed and abandoned house paint manufacturing plant.

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

To organize your data, first create a column labeled "Duplicate Frame Numbers" next to the original frame numbers Following that, 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 frame number Finally, adjust the formatting of the random numbers to display three decimal places for a polished appearance.

(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 20 sites that the field engineer should select for her study.

(g) Save the file as: RAND5

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 of individuals within 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, it would be impractical to survey every person in the U.S within that age range due to the extensive time and high costs involved in such a study.

To efficiently estimate the mean of an entire population, we utilize a sample of individuals rather than testing everyone, which conserves both time and resources This approach falls under "inferential statistics," as it involves inferring the population mean based on the sample mean.

In scientific research, analyzing a sample involves understanding its size (n), mean (X̄), and standard deviation (STDEV) These statistics are essential for estimating the population mean through a method known as the "confidence interval about the mean."

Estimating the Lower Limit and the Upper Limit

of the 95 % Confidence Interval About the Mean

This book does not delve into the theoretical background of the test; however, for a comprehensive understanding, readers are encouraged to consult reputable statistics textbooks such as McKillup and Dyar (2010) or Ledolter and Hogg (2010) The fundamental concepts underlying the test can be grasped through these resources.

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

In this book, we establish a "lower limit" and an "upper limit" for the population mean, aiming for a confidence level of 95% that the true mean lies within this interval.

“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.”

To highlight the Chevy Impala's lower environmental impact, we can confidently state that it achieves 28 miles per gallon (mpg), as this figure falls within the 95% confidence interval of our research, which ranges from 26.92 mpg to 29.42 mpg While the exact population mean remains unknown, we can assert that 28 mpg is a valid representation of the vehicle's fuel efficiency.

But we are only 95 % confident that the population mean is inside this interval, and

In scientific research, we accept a 5% margin of error when assuming that the population mean is 28 mpg, which allows us to maintain a 95% confidence level in our findings While this 95% confidence is somewhat arbitrary, it serves as a standard for our analysis throughout this book By consistently aiming for 95% confidence, we eliminate the need for readers to determine their own confidence levels for the problems presented Thus, we will always strive for 95% confidence in our results.

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, start with the sample mean (X̄) and add 1.96 times the standard error (s.e.) to determine the upper limit For the lower limit, subtract 1.96 times the standard error from the sample mean.

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).

36 3 Confidence Interval About the Mean Using the TINV Function and Hypothesis

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

To calculate the confidence interval, first add and subtract 1.96 times the standard error (s.e.) from the mean This process establishes the upper and lower limits of the confidence interval, where 1.96 s.e represents the product of 1.96 and the standard error of the mean.

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

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

Estimating the Confidence Interval for the Chevy

Impala in Miles Per Gallon

In a study examining the carbon footprint of Chevy Impala drivers, 49 owners recorded their mileage and fuel usage over two tanks of gas The average fuel efficiency was found to be 27.83 miles per gallon (mpg), with a standard deviation of 3.01 mpg Consequently, the standard error (s.e.) calculated 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 part of the ongoing 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 37

With the Chevy Impala achieving 28 mpg within the confidence interval, we can design a billboard that showcases its fuel efficiency while underscoring its reduced environmental impact Our data reinforces this assertion, making it an appealing choice for eco-conscious consumers.

28 mpg is inside of this 95 % confidence interval for the population mean.You are probably asking yourself: “Where did that 1.96 in the formula come from?”

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 the population data follows a "normal distribution," meaning that if we could test everyone or every property in the population, the data would resemble a "normal curve." This curve, which is symmetric like the outline of the Liberty Bell in Philadelphia, would perfectly align if folded in half, illustrating the balance of data around the mean.

Integral calculus is not the focus of this book; however, we aim to determine the lower and upper limits of population data within the normal curve, ensuring that 95% of the area lies 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 boundaries of our confidence interval For further exploration of this concept, refer to a reputable statistics book, such as McKillup and Dyar (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 to maintain a 95% confidence level in our results; therefore, whenever our research study involves more than 40 participants, we will consistently use a value of 1.96.

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

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, first determine the sample mean (X) For the upper limit, add the product of the t-value and the standard error (s.e.) to the sample mean Conversely, for the lower limit, subtract the product of the t-value and the standard error from the sample mean The t-value can be found 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 in Appendix E, labeled "sample size n," to determine the critical value of t necessary for your research study.

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 applicable for a 95% confidence interval around the mean For instance, with a sample size of 14 participants, the t value is 2.160.

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

If you have more than 40 people in your research study, the value of t is always 1.96.

In Appendix E, the "critical t column" provides the t-value necessary for achieving 95% confidence in your statistical test results This book operates under the assumption that you aim for 95% confidence, making the t-value from the t-table in Appendix E essential 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 from the appropriate statistical table Once you have the t-value, you can utilize Excel's functions to compute the confidence interval, ensuring accurate results for your data analysis.

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:ẳXTINV 1ð 0:95, n1ị*s:e:ðno spaces between these symbolsị ð3:3ị Upper limit:ẳXỵTINV 1ð 0:95, n1ị*s:e:ðno spaces between these symbolsị ð3:4ị

In Excel formulas, the “*” symbol indicates multiplication, representing the concept of "times" in mathematical terms Additionally, as mentioned in Chapter 1, "n" denotes the sample size, while "s" represents 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 determine the 95% confidence interval for the mean in a practical example.

Let’s suppose that General Motors wanted to claim that its Chevy Impala achieves 28 miles per gallon (mpg) Let’s call 28 mpg the “reference value” for this car.

As an employee at Ford Motor Co., you aim to validate 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 significant and reliable.

Using Excel to Find the 95 % Confidence Interval

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

A study involving a sample of new car owners was conducted to monitor their mileage over two tanks of gas, where participants recorded the average miles per gallon achieved The findings of this research are illustrated in Figure 3.1, showcasing the hypothetical results of the study.

To analyze the data effectively, create a spreadsheet in Excel and input the necessary values Utilize Excel functions to calculate the sample size (n), mean, standard deviation (STDEV), and standard error of the mean (s.e.) Ensure to reference the appropriate cells for accurate results.

Enter the other mpg data in cells A7: A30

To enhance the appearance of your table, first, select cells A6:A30 and format the numbers to one decimal place while centering them in Column A Next, increase the width of Columns A and B to twice their original size, and adjust Column C to be three times the width of the original Column A This formatting will give your table a more professional look.

Fig 3.1 Worksheet Data for Chevy Impala (Practical Example)

B26: Draw a picture below this confidence interval

B29: lower (right-align this word)

B30: limit (right-align this word)

C28: ‘ - 28 -–28.17 -– (note that you need to begin cell C28 with a single quotation mark (‘) to tell Excel that this is a label, and not a number)

D28: ‘ - (note the single quotation mark)

E28: ‘29.42 (note the single quotation mark)

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

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

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*D16ðno spaces between symbolsị

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

The confidence interval for the mean is calculated using the formula, which indicates a lower limit of 26.92 In this context, D10 represents the mean value, while D16 denotes the standard error of the mean An illustrative example of this confidence interval can be seen in Fig 3.3, highlighting the relationship between the mean and its associated confidence interval.

F23: ẳD10ỵTINV 1ð :95, 24ị*D16ðno spaces between symbolsị

The calculated upper limit of the confidence interval is 29.42 To ensure clarity in your Excel spreadsheet, format the mean, standard deviation, standard error of the mean, and both the lower (26.92) and upper limits of your confidence interval to two decimal places Currently, if you were to print this spreadsheet, the lower and upper limits would spill over onto a second page due to the excessive size of the displayed information.

To adjust the size of your Excel spreadsheet, utilize the "Scale to Fit" commands discussed in Chapter 2, Section 2.4, to reduce the spreadsheet to 95% of its current dimensions Access this feature through the Page Layout tab, and upon completion, observe the dotted line next to the measurements 26.92 and 29.42, which now indicates that these values will fit on a single printed page (refer to 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.

When creating a spreadsheet, it's important to format the conclusion across three separate lines instead of one long line This approach prevents two issues: first, printing the conclusion on a single line may result in an unreadable font size if the layout is reduced to fit on one page; second, if the layout is not adjusted, part of the conclusion could spill over onto a separate page, compromising the professional appearance of the spreadsheet.

The research study confirmed that the Chevy Impala achieved an average fuel efficiency of 28 miles per gallon, with the study's findings indicating a mean of 28.17 miles per gallon The resulting data has been saved in a spreadsheet titled CHEVY7.

Hypothesis Testing

Hypotheses Always Refer to the Population of Physical

Properties that You Are Studying

The first step is to understand that our hypotheses always refer to thepopulationof physical properties in a study.

In our study of vehicle headlight brightness, we will measure the lumens of various light bulb types used in these headlights By collecting brightness data from a diverse selection of bulbs, we aim to generalize our findings to represent all light bulbs utilized in this specific vehicle model.

In our study, we focus on the specific light bulbs used in this type of vehicle, which represent a sample from the larger population of all light bulbs utilized in similar vehicles.

Our sample sizes usually consist of only a fraction of the total light bulbs, so we focus on how the findings from our sample can be effectively generalized to the larger population we are studying.

That is why our hypotheses always refer to the population, and never to the sample of physical properties 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 the population 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)

These two hypotheses are called thenull hypothesisand theresearch hypothesis. Statistics textbooks typically refer to thenull hypothesiswith the notation:H0.

Theresearch hypothesisis typically referred to with the notation:H1, and it is sometimes called thealternative 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, the principle of "innocent until proven guilty" underpins the judicial process, where the null hypothesis assumes the defendant's innocence and the research hypothesis posits their guilt.

In Missouri, the state slogan "Show me" reflects the residents' skepticism towards unverified claims, emphasizing their belief that actions are more significant than words This mindset illustrates that Missourians prefer tangible evidence over mere assertions, highlighting their demand for proof in communication.

In hypothesis testing, the goal is to determine which of the two competing statements—the null hypothesis or the research hypothesis—should be accepted as true, as both cannot coexist simultaneously By applying statistical formulas, researchers evaluate the evidence to decide which hypothesis to reject.

In scientific research, rating scales are frequently employed to assess individuals' attitudes towards a company, its products, or their purchasing intentions Commonly used scales include 5-point, 7-point, and 10-point formats, although various other scale values may also be utilized.

3.2.2.1 Determining the Null Hypothesis and the Research Hypothesis

When Rating Scales Are Used

This article explores alternative methods for testing null and research hypotheses through rating scales, which are rarely utilized in engineering sciences It presents a practical example of a 7-point scale that the American Society of Mechanical Engineers (ASME) might employ to gather participant feedback on the value of its annual International Mechanical Engineering Congress and Exposition technical conference.

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:

According to our statistical test results for the attitude scale item, if the population mean is approximately 4, we conclude that we accept the null hypothesis, indicating that ASME conference participants felt neutral regarding their satisfaction with the conference quality.

If our statistical test reveals a significant difference between the population mean and the value of 4, we reject the null hypothesis in favor of the research hypothesis.

“ASME conference participants were significantly satisfied with the quality of the conference” (this is true whenever our sample mean is significantly greater than our expected population mean of 4). or

Participants of the ASME conference expressed considerable dissatisfaction with the event's quality, as evidenced by a sample mean that falls significantly below the anticipated population mean of 4.

Fig 3.6 Example of a Rating Scale Item for a Conference (Practical Example)

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.

The research engineer's role involves determining whether to accept the null hypothesis or the research hypothesis as true based on the data collected in the 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:

Here are the answers to these three questions:

1 The null hypothesis is μ ẳ 3, and the research hypothesis is μ 6ẳ 3 on this 5-point scale (i.e the “middle” of the scale is 3).

2 The null hypothesis is μ ẳ 4, and the research hypothesis is μ 6ẳ 4 on this 7-point scale (i.e., the “middle” of the scale is 4).

3 The null hypothesis isμ ẳ 5:5, and the research hypothesis isμ 6ẳ 5:5 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).

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

Webster University, located in St Louis, Missouri, implements a Course Feedback form for student evaluations at the conclusion of its pre-engineering courses This form includes 12 rating items that assess the course's planning, organization, and instructor-student communication After the course ends, the ratings are summarized, and the results are provided to instructors, utilizing a 4-point scale for evaluations.

In this analysis, the null hypothesis posits that the mean (μ) is equal to 2.5, while the research hypothesis suggests that the mean is not equal to 2.5 This is based on the observation that there are two ratings below 2.5 and two ratings above 2.5 on the scale, which is designed so that lower scores indicate better outcomes, similar to scoring in golf.

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

the Confidence Interval About the Mean

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 is crucial to focus on the midpoint of the scale For instance, in a 7-point scale where 1 represents "poor" and 7 signifies "excellent," the hypotheses should center around the average responses, which reflect the central tendency of the data collected.

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 process for calculating the confidence interval for the mean using Excel The key steps involved in applying this formula were thoroughly discussed.

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 (We Will Explain Step 4 Later in the Chapter.)

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 utilizing the confidence interval for the mean, there are two potential outcomes, but only one can be deemed "true." Therefore, your findings will align with one of these results.

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 Test in Plain English!

Summarizing the results of a statistical test in straightforward language can be challenging, especially when aiming for clarity for those without a statistics background, such as a supervisor This book will provide ample practice to help you effectively communicate the conclusions of your confidence interval analysis regarding the mean test, ensuring that your findings are both concise and easily understood.

Let’s set some basic rules for stating 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:

The reference value of 28 mpg falls within the 95% confidence interval for the data analyzed, 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, the term "significantly" cannot be used in the conclusion This guideline is consistently applied to 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 straightforward, as it reflects the statements made in the null hypothesis In contrast, formulating a conclusion when rejecting H0 and accepting the alternative hypothesis (H1) is more complex To enhance your skills in this area, 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 recorded was only 31 mpg Additionally, it is important to clarify that simply stating a result as "significantly less than" the null hypothesis is not adequate; further context is necessary to fully understand the implications of the findings.

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.

The density of a substance remains consistent regardless of the quantity, making it a crucial factor in mineral identification Density is determined by dividing the mass (g) of an object by its volume (cm³) For instance, pure silver has a density of 10.49 g/cm³; if a sample's density differs from this value, it cannot be classified as pure silver, regardless of its appearance Variations in density can occur due to impurities or mixtures, underscoring the reliability of density measurements in scientific analysis.

To assess the purity of a substance claimed to be pure silver, we analyzed 50 random samples from the company's purchase by calculating their densities The resulting data allowed us to generate a confidence interval, which is crucial for determining whether the material meets the standard density of pure silver This analysis not only enhances our data interpretation skills but also provides valuable insights into the authenticity of the purchased material.

The hypotheses for this test would be:

The null hypothesis posits that a mean density of 10.49 g/cm³ indicates the substance purchased by the company is pure silver If the sample's mean score does not significantly differ from this value, it supports the conclusion of purity.

Suppose that your analysis produced the following confidence interval for this test:

10.41 _10.43 _10.45 10.49 _ 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.

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

– less or greater (depending on your result)

– either pure silver or not pure silver (since the result is significant)

The density of the tested substance was found to be significantly lower than that of pure silver, suggesting it was likely around 10.43 g/cm³ Consequently, this indicates that the substance in question is not pure silver.

Note that you need to use the word “less” since the sample mean of 10.43 g/cm 3 was less than the reference value of 10.49 g/cm 3

As a quality control supervisor at a machine shop known for producing high-quality steel rods for construction, you are tasked with verifying the accuracy of a recently repaired cutting machine The machine, which had been experiencing issues with precise cuts, has undergone fixes, and management seeks confirmation of its performance Utilizing your Excel skills, you will analyze a random set of test rods that were intended to be cut to a length of 5.5 cm to ensure that the machine is now functioning correctly and meeting the established standards.

Suppose that your research produced the following confidence interval for this machine for your test:

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

– longer or shorter (depending on the result of your test)

Rule #3: The sample of test bars were cut significantly longer than what the cutting machine was set for at 5.5 cm, and were probably closer to 5.8 cm.

In conclusion, it's essential to note that while native English speakers may not typically use the phrase "significantly longer," your statistical evidence lends credibility to your statement Additionally, the mean measurement of 5.8 cm exceeds the reference value of 5.5 cm, highlighting a notable difference.

If you want a more detailed explanation of the confidence interval about the mean, see Townend (2002).

Alternative Ways to Summarize the Result

The 7 STEPS for Hypothesis-Testing Using

The 9 STEPS for Hypothesis-Testing Using

Tiêu đề	Excel 2016 for Engineering Statistics: A Guide to Solving Practical Problems
Tác giả	T.J. Quirk, M. Quirk, H.F. Horton, E. Rhiney, S. Cummings, J. Palmer-Schuyler
Người hướng dẫn	Thomas J. Quirk
Trường học	Springer International Publishing
Chuyên ngành	Engineering Statistics
Thể loại	guide
Năm xuất bản	2016
Thành phố	Switzerland

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Số trang	258
Dung lượng	11,54 MB