May Takeuchi, Alex Takeuchi
April 2, 2019
... questionnaire to customers at Bubba’s Southern Burger asking their basic demographic background and how often they have had Bubba’s most expensive “Beefy Double Patty Magnum Burger ” known for its “meatiness” during the last 30 days. From all the respondents, you randomly select 15 customers whose age falls between 22 and 37 (“Millennials”) and another 15 customers whose age falls between 55 and 70 (“Baby boomers”), then compared how often they have had this particular burger in question. Below is the result:
Millennials
Baby boomers
2
4
7
12
8
2
5
4
2
10
1
9
2
8
4
6
6
8
7
3
1
4
2
6
4
3
3
4
1
4
Q1: State the null hypothesis using appropriate symbols. (*Hint: Which t-test would be appropriate in this study?)
Example Answer: H0: µM = µB (Or H0: µM - µB = 0)
Q2: Using Excel, examine if Millennials and Baby boomers are significantly different in their preferences for “meaty” double-patty burger at the alpha level of .05. Report the result in the APA format.
Example Answer: t(28), p < .05, two-tailed. Reject H0.
Q3: Based on this small study, would you consider either of the age cohorts is more or less meat-eaters than the other? What is your overall conclusion?
Example answer: The difference between the two cohorts in their appitite for a beefy double-patty burger was statistically significant. Baby boomers are found to be more meat-eaters than Millennials.
Q4: During the study, you found that most respondents in the Millennials group were students while most of in the Baby Boomers group were professors who were having lunch at the burger restaurant. Using the additional information on the demographic characteristics of the respondents, discuss if there is any factor other than the independent variable (i.e., age cohorts) that may be influencing the observed result of the study. In other words, speculate if it is possible to explain the observed difference between the two groups (i.e., frequency of eating the Bubba’s most expensive Beefy Double Patty Magnum Burger ) by some other sociological variable than “age cohorts.”
Example answer 1: Professors can afford expensive burgers more often than students can because their income level is significantly higher than that of students. The difference may simply be a product of income difference but not of age cohorts or associated sub-culture per se.
Example answer 2: There seem to be more male professors than female professors in the particular age cohort “Baby boomer.” On the other hand, there seem to be more female college students than male students nowadays. If men are more likely to eat “meaty” burgers than women, our “Baby Boomer” respondents, that include more males, would naturally be more “meat eaters” than our “Millennial” respondents that include more females.
Example answer 3: Students eat lunch at various places, both on- and off-campus, while professors are more likely to go off-campus to have lunch because they have more time and can afford (?) The study may simply be showing professors’ stronger tendency to have lunch at an off-campus fast food restaurant and have ANY burgers including the meaty burger, while students may be showing their general tendency to have lunch on-campus (thus to less frequently eat “any burger” off-campus). Therefore, frequency of eating the meatiest burger at an off-campus restaurant is not a valid operationalization of their preference for “meaty food,” anyway. Hamburger Statistics 14
Hamburger Statistics – Introduction to T-Tests
A. Independent Samples t-Test: Comparing Two Unrelated Groups
Your task: To examine if two populations differ in some important way by taking a sample from each population and then analyzing the difference between the two samples.
Scenario: There are two very popular hamburger joints nearby campus: Pacific Northwest Burger (simply “PNW” hereafter) that offers stylish and zesty hamburgers with modern twists and Bubba’s Southern Burger (simply “Bubba’s” hereafter) that offers good old school rich and meaty hamburgers. Both restaurants offer comparable all American hamburgers practically for the same prices. But die-hard supporters of PNW claim that their burgers are more refined, tasty and satisfying; while those who swear by Bubba’s claim that their burgers are more authentic, delicious and satisfying. Yet, quite a few say that when it comes to satisfaction it is hard to rank them because both restaurants’ burgers are delicious and equally satisfying, despite having different emphases.
There is plenty of anecdotal evidence floating around to support either side, but no one seems to have reliable empirical evidence to validate his/her claim… So, which restaurant’s burgers are generally more satisfying over all? Or are they equally satisfying? As an unbiased sociology major who frequent both restaurants equally often, you are asked to conduct a simple survey study and find out the answer to the question with evidence.
1. Example: You have randomly selected 10 students who are regulally eating lunch at PNW, then asked how satisfied they are generally with PNW’s burgers on a scale of 0 to 100, where 0 is no satisfaction and 100 is the highest level of satisfaction. You have done the same for those students who are regularlly eating luch at Bubba’s. The table below shows the ratings given by 10 students from each restaurant, a total of 20 students:
PNW
Score
Bubba’s
Score
Student 1
67
Student 11
87
Student 2
74
Student 12
95
Student 3
73
Student 13
96
Student 4
94
Student 14
95
Student 5
69
Student 15
86
Student 6
86
Student 16
75
Student 7
96
Student 17
85
Student 8
73
Student 18
68
Student 9
68
Student 19
68
Student 10
80
Student 20
72
In this study, you are examining if there is statistically significant difference between the population means for the PNW patrons and for the Bubba’s patrons in terms of their overall satisfaction levels. Therefore, using proper symbols, the null hypothesis is stated as:
H0: μP = μB.
2. Testing the null hypothesis: Procedures
a. Open Excel for a clean spreadsheet. For headings, type “PNW” in cell A1, “Bubba’s” in cell C1, “Sample” (for “sample number”) in cells A2 and C2, and “Score” in cells B2 and D2.
b. Starting at cell A3, enter the numbers from 1 to 10 for all 10 samples from “PNW” in the column for “Sample.” Starting at cell C3, enter the numbers from 11 to 20 for all 10 samples from “Bubba’s” in the column for “Sample.”
c. Starting at cell B3, enter the specific score for each of the 10 samples from “PNW” in the column for “Score.” Starting at cell D3, enter the specific score for each of the 10 samples from “Bubba’s” in the column for “Score.”
d. Save the file as “Hamburger Stats_ttests.xlsx.”
e. Type “n” in cells A13 and C13, “Average” in cells A14 and C14, “StdDev” in cells A15 and C15, and “ttest” in cell A16. Does your spreadsheet look like below?
f. Move the cursor over to and select cell B13, and then click on the “fx” symbol in the toolbar at the top. When Formula Builder appears, select “COUNT” in the dropdown menu, and then click on “Insert Function.”
g. Enter B3:B12 in Value1 box. Hit Done. This gives you the sample size (i.e., n) for PNW. Repeat the procedure for Bubba’s to find its sample size in cell D13.
h. Move the cursor over to and select cell B14, and then click on the “fx” symbol in the toolbar at the top. When Formula Builder appears, select “AVERAGE” in the dropdown menu, then click on “Insert Function”.
i. Enter B3:B12 in Number1 box. (Caution: Excel may be auto filling the reference rage by including B13. To clear this, double-check your entries in the formula!) Hit Done. This gives you average (i.e., P) for PNW. Repeat the procedure for Bubba’s to find its average (i.e., B) in cell D14.
Did you get the results for PNWand 82.7 for Bubba’s?
Should you conclude that overall Bubba’s burgers are actually more satisfying than PNW’s? Or, is the 4.7 points difference just a matter of margin of error?
j. Compute the Standard Deviation for PNW by 1) moving the cursor over to and select cell B15; 2) clicking on the “fx” symbol in the toolbar at the top; 3) selecting “STDEV” in the dropdown menu; 4) clicking on “Insert Function;” and 5) entering B3:B12 in Number1 box in Formula Builder. (Again, Excel may be auto filling the reference rage by including B13 and B14. To clear this, double-check your entries in the formula!) Then, hit “Done.” Repeat the procedure for Bubba’s to find its Standard Deviation in cell D15.
Compare and contrast the two standard deviations for the restaurants. Are the two standard deviations similar to or very different from each other?
k. Move the cursor over to and select cell B16, and then click on the “fx” symbol in the toolbar at the top. When Formula Builder appears, select “TTEST” in the dropdown menu, then click on “Insert Function”.
l. Enter B3:B12 in “Array1,” D3:D12 in “Array2,” 2 in “Tails,” and then 2 in “Type” if the two standard deviations (i.e., the values in B15 and C15) can reasonably be treated as equal to each other. If they are unequal (i.e., one is larger than twice the size of the other), enter 3 in “Type.” Hit “Done.”
m. The value in cell B16 indicates the probability of obtaining the mean difference between PNW and Bubba’s that falls in the two tails. See if the value is higher or lower than α = .05 (i.e., 5%).
If the t-statistic value is smaller than .05, it means that the probability of obtaining the result by chance is less than .05 (5%) or “very unlikely.” So the difference is “statistically significant.”
If the t-statistic value is greater than .05, it means that the probability of obtaining the result by chance is more than .05 (5%) or “NOT very unlikely” (which means “possible”). So the difference is “NOT statistically significant.”
n. Report the result in the APA format.
Template: t(df), p > .05, two-tailed. Reject/Fail to reject H0
df = degrees of freedom = (n1- 1) + (n2 - 1)
In this example question, nP of PNW and nB of Bubba’s are both 10. Therefore, df = (10-1) + (10-1) = 18.
< if the obtained t-statistic value is smaller than .05; then state “Reject H0.”
> if the obtained t-statistic value is greater than .05; then state “Fail to reject H0”
For the analysis in this example, you may report the result and state the conclusion as follows:
t(18), p > .05, two-tailed. Fail to reject H0. There is no statistically significant difference in the level of overll satisfaction between the patrons of Pacific Northwest Burger and the patrons of Bubba’s Southern Burger. Therefore, there is no evidence to validate that either restaunt’s burger is more satisfying than the other.
3. Exercise: Both PNW and Bubba’s also offer basic milkshakes for the same price. How about their strawberry milkshakes then? Are the strawberry shakes from the two restaurants equally good or is one better than the other?
You have randomly selected 15 regulars at PNW who have had a strawberry milkshake there, then asked how they would rate it on the scale of 0 to 100, where 0 is the lowest and 100 is the highest. You have done the same with regulars at Bubba’s. The table below shows the ratings given by a total of 30 customers:
PNW
Score
Bubba’s
Score
Customer 1
75
Customer 16
86
Customer 2
76
Customer 17
80
Customer 3
77
Customer 18
85
Customer 4
72
Customer 19
74
Customer 5
65
Customer 20
75
Customer 6
78
Customer 21
83
Customer 7
77
Customer 22
85
Customer 8
70
Customer 23
88
Customer 9
82
Customer 24
88
Customer 10
75
Customer 25
86
Customer 11
83
Customer 26
78
Customer 12
81
Customer 27
90
Customer 13
87
Customer 28
72
Customer 14
85
Customer 29
75
Customer 15
79
Customer 30
85
Question a: State the null hypothesis using appropriate symbols.
Question b: Using the data provided above, examine if the strawberry shake from one restaurant tastes significantly better (or worse) than the other at α = .05.
B. Dependent Samples t-Test: Comparing Two Related Groups
Your task: To examine if there is any difference between two conditions (e.g., treatment 1 vs. treatment 2 or “before” vs. “after”) within a population by repeating observations on a single sample.
Scenario: The big campus wide debate over Pacific Northwest Burger vs. Bubba’s Southern Burger still continues... To study which restaurant’s signature burger students like better, you decided to adopt a different method this time as the two restaurants agreed to provide their signature burgers for this study: To have a same group of students taste and compare the signature burgers from the two restaurants.
1. Example: You have randomly selected 8 new students who have not eaten at either restaurant yet, and gave each student a half portion of two burgers for lunch, one from PNW’s signature burger Cascade Excellence and another from Bubba’s signature burger Smoky Mountains Finest . Then, you asked how they would rate each burger on the scale of 0 to 100. The table below shows the ratings given by a total of 8 students:
Student
Cascade
Smoky
1
75
85
2
80
90
3
70
85
4
80
75
5
75
80
6
75
65
7
85
80
8
90
95
In this study, you are examining if there is statistically significant mean difference in the population between the paired scores of Cascade Excellence and Smoky Mountains Finest. Therefore, using symbols, the null hypothesis is stated that:
H0: μd = 0
2. Testing the null hypothesis: Procedures
a. On “Hamburger Stats_ttests.xlsx,” open a new spreadsheet by clicking on the + sign at the bottom.
b. For headings, type “Sample” in cell A1, “Cascade” in cell B1, and “Smoky” in cell C1. Starting at cell A2, enter the number 1 to 8 for the eight students who participated in this study. Enter “n of pairs” in Cell A10, then enter the number of paired observations (i.e., the number of students who participated in the study) in this study in Cell B10.
c. Enter “Average” in cell A11, “StdDev” in cell A12, and “ttest” in cell A13.
Starting at cell B2, enter the scores for “Cascade” above in column B on the spreadsheet. Starting at cell C2, enter the scores for “Smoky” in column C.
d. Using Function (fx), compute the mean average for each of the two, Cascade ( and Smoky (), in Cell B11 and C11 repectively.
Did you get 78.75 and 81.88?
e. Should we conclude that Smoky Mountains Finest, the signature burger of Bubba’s actually tastes better than Cascade Excellence, the signature burger of PNW? Or, is the 3.13 pts difference (i.e., 81.88 pts – 78.75 pts) just a matter of margin of error? Let’s find out!
f. Compute the standard deviation for each group in B12 and C12.
g. Place the cursor in B13. Move the cursor to the fx button and click on it. In Formula Builder, select “TTEST,” then hit “OK.” Enter B2:B9 in “Array1,” C2:C9 in “Array2,” 2 in “Tails,” and then 1 in “Type.” Hit “Done.”
h. The value in B13 indicates the probability of obtaining different mean scores for Cascade and Smoky to fall in the two tails. See if the value is higher/lower than α = .05 (i.e., 5%).
i. State your conclusion in the APA format.
Template: t(df), p > .05, two-tailed. Reject/Fail to reject H0
df = degrees of freedom = n-1= (number of pairs) – 1 (*In this example question, n = 8. Therefore, df = 7.)
3. Exercise:
Scenario: Suppose that your friend who works at Bubba’s came to you with a concern. She was a little worried about the taste of French fries at Bubba’s. “Our French fries sales are dropping recently, especially after Pacific Northwest Burger started offering their new Pacific Sea-Salted Fries last month. Do their new fries actually taste better than ours? Can you help me check that out?”
“Sure!” you went. “I have just learned t-test in my research methods class last week. I can examine if the new fries from Pacific Northwest Burger in fact tastes better than those from Bubba’s Southern Burger or they are about the same.”
You have selected a random sample of 16 students on campus and asked them to blindly rate the taste of French fries from the two burger restaurants on the 0 – 100 point scale. Of the 16 students, 8 were asked to rate Pacific Sea-Salted Fries from PNW and another 8 were asked to rate Bubba’s. Below is the result:
PNW
Score
Bubba’s
Score
Student 1
80
Student 9
70
Student 2
75
Student 10
60
Student 3
80
Student 11
70
Student 4
95
Student 12
85
Student 5
90
Student 13
75
Student 6
80
Student 14
80
Student 7
85
Student 15
85
Student 8
90
Student 16
40
Question a: State the null hypothesis using appropriate symbols. (*Hint: Which t-test are you performing, independent t-test or dependent t-test?)
Question b: Using Excel, examine if the new French fries from PNW taste significantly better than those from Bubba’s at α = .05. State the result in the APA format.
Question c: Given this result, should your friend be concerned about the taste of Bubba’s fries? What is your conclusion?
Scenario (ctnd.): Having learned that their French fries are indeed not as popular as PNW’s new fries, your friend working at Bubba’s Burger has been working hard to improve the taste of their French fries ….
When you stop by at Bubba’s a few weeks later, your friend looks really excited, saying “Hey, I’ve been fixin’ to git new French fries for some time. Now, I have them! I sprinkled special Southern BBQ rub on our fries, and they sure taste better than ever! These are called “Dixie Fries.” Try some!”
Do the new Dixie Fries from Bubba’s taste significantly better than their original French fries? To find out, you have selected a random sample of 10 students on campus; and then asked them to eat a small portion of the new Dixie Fries first, and then the same amount of the original French fries. The 10 subjects rated each version of fries on the 0 – 100 point scale. Below is the result:
Dixie Fries
Original fries
Student 1
80
65
Student 2
90
70
Student 3
80
85
Student 4
90
70
Student 5
90
80
Student 6
85
80
Student 7
80
80
Student 8
90
70
Student 9
85
80
Student 10
80
70
Question d: State the null hypothesis using appropriate symbols.
Question e: Using Excel, examine if the new Dixie Fries taste significantly better than the original fries at α = .05. Report the result in the APA format. Finally, state your overall conclusion.
Question f: Some of your subjects may have already been full by the time they had the second pack of French fries; and thus the order in which they ate those two versions of French fries could be affecting the subjects’ ratings (i.e., the first pack of French fries may taste better to the subjects simply because they eat them before they have become satiated.) What would you do to solve this problem? Name the methodological solution that you learned in class, and briefly explain how it will work.
C. Applied Questions: Which age cohort is more meat-eaters: Millennials or Baby boomers?
Scenario: You are assigned to a small gerontological research project to investigate an effect of aging on some everyday social behavior. Having had several studies on the two burger restaurants, you have now come to wonder if age cohort has anything to do with their food preferences. “Millennials are more self-conscious about their eating habits and thus may not be craving for red meat so much; however, they may need more protein and calories to support their more active lifestyles than the baby boomers. On the other hand, while baby boomers would probably focus more on healthy diets and balanced nutrition to prevent geriatric illnesses, they may find traditional “meaty” burgers with big juicy patties more attractive and satisfying because that is what they grew up with….” Would there be any difference in their eating tendency towards big meaty burgers at all? In short which cohort of people would be eating “beefy,” “juicy,” “meaty,” burgers more often?
To test the hypothesis, you distribute a survey questionnaire to customers at Bubba’s Southern Burger asking their basic demographic background and how often they have had Bubba’s most expensive “Beefy Double Patty Magnum Burger ” known for its “meatiness” during the last 30 days. From all the respondents, you randomly select 15 customers whose age falls between 22 and 37 (“Millennials”) and another 15 customers whose age falls between 55 and 70 (“Baby boomers”), then compared how often they have had this particular burger in question. Below is the result:
Millennials
Baby boomers
2
4
7
12
8
2
5
4
2
10
1
9
2
8
4
6
6
8
7
3
1
4
2
6
4
3
3
4
1
4
Q1: State the null hypothesis using appropriate symbols. (*Hint: Which t-test would be appropriate in this study?)
Q2: Using Excel, examine if Millennials and Baby boomers are significantly different in their preferences for “meaty” double-patty burger at the alpha level of .05. Report the result in the APA format.
Q3: Based on this small study, would you consider either of the age cohorts is more or less meat-eaters than the other? What is your overall conclusion?
Q4: During the study, you found that most respondents in the Millennials group were students while most of in the Baby Boomers group were professors who were having lunch at the burger restaurant. Using the additional information on the demographic characteristics of the respondents, discuss if there is any factor other than the independent variable (i.e., age cohorts) that may be influencing the observed result of the study. In other words, speculate if it is possible to explain the observed difference between the two groups (i.e., frequency of eating the Bubba’s most expensive Beefy Double Patty Magnum Burger ) by some other sociological variable than “age cohorts.” Hamburger Statistics – Introduction to T-Tests
Usage notes
1. This resource was originally developed while tutoring students with mild math and stats anxiety. It can be used to ease students’ “psychological barriers” to statistics and to introduce them to the most basic statistical concepts before they take on more advanced and sociologically inclined questions such as those given as exercises in standard statistics and research methods textbooks in Sociology.
2. Because of the fundamental nature of the statistical concepts covered herein, this recourse can be adopted by instructors of introductory social science courses as well as those teaching research methods and statistics.
3. This resource would be especially suitable for teaching elementary statistics in an online format. The step-by-step instructions on the use of Excel can easily be converted to Power Point slides, and the assignment questions can be presented on the screen in sequence.
4. One common mistake that students tend to make on this assignment is entering a wrong reference range in the formula in Excel. For example, to calculate the average for cells A2 through A6, they may enter “AVERAGE(A2:A7)” instead of “AVERAGE(A2:A6).” This error occurs when Excel auto-fills the reference range where A7 displays some other value such as n (i.e., the number of entries in cells A2 through A6). However, such an error can easily be avoided by instructing students to double-check their entries.
Other than that, the authors have observed no recurrent problems or common errors made by students on this assignment. Because it is a very simple and fun exercise, students generally respond to the instructions and questions very well without confusion or “anxiety.” ..."
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Subject Area(s):
- Statistics
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Resource Type(s):
- Assignment
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Class Level(s):
- College 300
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Class Size(s):
- Any
- Abstract:
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Teaching materials on quantitative analysis are often stereotyped as "dry," "boring," "difficult" or even "intimidating" by some undergraduate students. To make the entry level inferential statistics more approachable to those students, this assignment uses...