4.8 Paired T-test


Use a paired t-test when each subject has a pair of measurements, such as before and after scores. A paired t-test determines whether the mean change for these pairs is significantly different from zero.


In StatCrunch enter sample data in two columns, then select Stat → T Stats → Paired



  1. You are a bean sorter in a bean-sorting facility. Your pay is dependent on how many beans you can sort during your shift. One day, evaluators from OSHA visit the facility and discover that you are sorting beans without wearing gloves!
    1. Do you think wearing gloves will make a difference in your bean-sorting speed? How?


    2. Claim: Workers will sort beans slower when wearing gloves.
    3. \(H_0\): \(\mu_D=0\)
    4. \(H_A\): \(\mu_D < 0\) if subtract without gloves minus with gloves
    5. How many beans did you sort in the allotted time while wearing gloves?
    6. How many beans did you sort in the allotted time while not wearing gloves?
    7. Compile your classroom data and run a paired t-test using a significance level of 0.05. What is the p-value?
    8. Decision about the null hypothesis:
    9. Concluding statement about your claim:
    10. Should you approach your boss with a request for a pay rate increase now that you have to wear gloves? Why or why not?

  2. A researcher conducts an investigation to see how effective a new diet is in lowering cholesterol. Cholesterol level is measured before and after participants follow a special diet for 3 months. Are the subjects’ cholesterol levels lower on average after the diet? Test at the 0.05 significance level.
    Subject A B C D E F G H I
    Before 209 210 205 198 216 217 238 240 222
    After 199 207 189 209 217 202 211 223 201

    1. Claim: Cholesterol levels are lower after the special diet.
    2. \(H_0\): \(\mu_D=0\)
    3. \(H_A\): \(\mu_D > 0\)
    4. What is the p-value in the paired t-test for the researcher’s claim? 0.013
    5. Decision about the null hypothesis? Reject \(H_0\)
    6. f. What should the researcher conclude about the new diet’s effectiveness in lowering cholesterol? There is sufficient sample evidence to support the claim that cholesterol levels are lower after the special diet.
    7. Construct a 95% confidence interval for the mean change in cholesterol level. (1.7, 19.9)
    8. What does the confidence interval tell us about the average change in cholesterol level for people following the special diet for 3 months? We are 95% confident that the mean change in cholesterol levels for people on the special diet for 3 months is between 1.7 and 19.9 lower.
    9. After the special diet, subject G’s cholesterol level was 27 points lower. Would it be prudent for an advertisement to say that this special diet will lower your cholesterol level by at least 20 points? Why or why not? It would not be advisable since the upper limit of the confidence interval for the mean change in cholesterol level is only 19.9 points.

  3. Five ball players think they can throw just as far with their off-hand (catching hand) as their dominant hand (throwing hand). Data was collected for the number of feet players threw the ball with each hand. Conduct a hypothesis test to determine whether the mean difference in distances is significant. Test at the 5% level.

    Player 1 Player 2 Player 3 Player 4 Player 5
    Dominant Hand 120 111 135 140 125
    Off hand 105 109 98 111 99

    1. Claim: Players can throw the same distance with their dominant hand and off hand.
    2. \(H_0\): \(\mu_D=0\)
    3. \(H_A\): \(\mu_D \neq 0\)
    4. What is the p-value in the paired t-test for the researcher’s claim? 0.023
    5. Decision about the null hypothesis? Reject \(H_0\)
    6. What should the researcher conclude about the new diet’s effectiveness in lowering cholesterol? There is sufficient sample evidence to reject the claim that players can throw the same distance with their dominant hand and off hand.

  4. Students in a statistics class thought they could do better on an exam if they were allowed to refer to their class notes during the exam. The teacher gave two exams on the same unit. On one exam the students got to use notes and on the other they didn’t. Students were randomly assigned to the exam where they got to use notes. Conduct a hypothesis test using a 0.10 significance level to determine whether students did better on an exam when they were allowed to use class notes.
    Student Exam score using notes Exam score without using notes
    Liam 85 83
    Olivia 77 80
    Noah 93 90
    Emma 78 72
    Mateo 84 75
    Mia 73 73
    William 89 85
    Luna 65 71
    Lucas 89 95
    Evelyn 76 77
    1. Claim: Statistics students do better on exams when they can refer to class notes during the exam.
    2. \(H_0\): \(\mu_D=0\)
    3. \(H_A\): \(\mu_D > 0\)
    4. What is the p-value in the paired t-test for the researcher’s claim? 0.3112
    5. Decision about the null hypothesis? Fail to Reject \(H_0\)
    6. What should the researcher conclude about the new diet’s effectiveness in lowering cholesterol? There is not sufficient sample evidence to support the claim that statistics students do better on exams when they can refer to class notes during the exam.

    5. Hypothesis Testing Decisions


    Put a box around the claim in the problem. Circle the little words in each problem that have big meaning!



  5. Problem

    Type of Problem

    Evidence

    You believe PSCC students on average watch less television now then during the covid pandemic. You randomly survey 90 PSCC students and ask them how many hours of television they watch per day. The mean number of hours in your sample is 3.5, with a standard deviation of 2.2. Can you support the claim that PSCC students watch less than 4 hours of television per day on average?
    • Hypothesis test for a population proportion – use Proportion Stats

    • Hypothesis test for a population mean – use T Stats

    • Hypothesis test for linear correlation – use Regression

    • Hypothesis test for Independence – use Tables ~ Contingency

    • Hypothesis test for mean change – use T Stats ~ Paired

    How did you know the type of hypothesis test?




    Claim:



    \(H_0\)


    \(H_A\)



  6. Problem

    Type of Problem

    Evidence

    A new prep class was designed to improve ACT scores. The scores on two practice exams for several randomly selected students were recorded, one before the class and one after the class. Are the scores, on average, higher after the class?
    Student Pre Post
    1 17 19
    2 21 22
    2 22 25
    2 16 20
    2 18 16
    • Hypothesis test for a population proportion – use Proportion Stats

    • Hypothesis test for a population mean – use T Stats

    • Hypothesis test for linear correlation – use Regression

    • Hypothesis test for Independence – use Tables ~ Contingency

    • Hypothesis test for mean change – use T Stats ~ Paired

    How did you know the type of hypothesis test?




    Claim:



    \(H_0\)


    \(H_A\)



  7. Problem

    Type of Problem

    Evidence

    In a random sample of 200 Transit Railroads passengers, the travel distance and type of ticket is recorded. The railroad wants to know if a passenger’s choice in ticket class is dependent on the distance they must travel.
    Miles 3rd Class 2nd Class 1st Class
    1-100 21 14 6
    101-200 18 16 8
    201-300 16 17 15
    301-400 12 14 21
    401-500 6 6 10
    • Hypothesis test for a population proportion – use Proportion Stats

    • Hypothesis test for a population mean – use T Stats

    • Hypothesis test for linear correlation – use Regression

    • Hypothesis test for Independence – use Tables ~ Contingency

    • Hypothesis test for mean change – use T Stats ~ Paired

    How did you know the type of hypothesis test?




    Claim:



    \(H_0\)


    \(H_A\)



  8. Problem

    Type of Problem

    Evidence

    About 20% of Americans over age 5 speak a language other than English at home. A politician claims that less than 20% of Knoxville residents speak a language other than English at home. The politician’s campaign randomly surveyed 430 Knoxville residents and found that 80 of them spoke a language other than English at home. Does the sample data support the politician’s claim?
    • Hypothesis test for a population proportion – use Proportion Stats

    • Hypothesis test for a population mean – use T Stats

    • Hypothesis test for linear correlation – use Regression

    • Hypothesis test for Independence – use Tables ~ Contingency

    • Hypothesis test for mean change – use T Stats ~ Paired

    How did you know the type of hypothesis test?




    Claim:



    \(H_0\)


    \(H_A\)



  9. Problem

    Type of Problem

    Evidence

    The CEO of an online store company claims there is no correlation between advertising costs and sales. Test the CEO’s claim using the sample data from the past year for 7 online stores.
    Advertising Dollars (in 1000s) E-Commerce Sales (in 1000s)
    1.7 368
    1.5 340
    2.8 665
    5 954
    1.3 331
    2.2 556
    1.3 376
    • Hypothesis test for a population proportion – use Proportion Stats

    • Hypothesis test for a population mean – use T Stats

    • Hypothesis test for linear correlation – use Regression

    • Hypothesis test for Independence – use Tables ~ Contingency

    • Hypothesis test for mean change – use T Stats ~ Paired

    How did you know the type of hypothesis test?




    Claim:



    \(H_0\)


    \(H_A\)