Stat 219

Study Guide-Week 7

Section 5.2-

Two Sample Test for Paired Data

(1) Identify whether each of the following examples is an example of a paired dataset:

a)Consider a dataset with the ages of 25 married women on a staff in one column, and the ages of 30 married men on the staff of a particular organization in the second column, is this considered a paired dataset? Yes/No

b)Consider a dataset with the ages of 25 married women on a staff in one column and the ages of 25 married men on the staff in the second column of a particular organization, is this considered a paired dataset? Yes/No

c)Consider a dataset with the ages of 25 married women on a staff in one column and the ages of their husbands in the second column in a random order, (husbands not necessarily listed next to their wives) is this considered a paired dataset? Yes/No

d)Consider a dataset with the ages of 25 married women on a staff in one column and the ages of their husbands in the second column (with the husbands' age listed next to each woman staff member)is this considered a paired dataset? Yes/No

e)Compare pre- (beginning of semester) and post-test (end of semester) scores of students. Yes/No

f)Assess gender-related salary gap by comparing salaries of randomly sampled men and women. Yes/No

g)Compare artery thicknesses at the beginning of a study and after 2 years of taking Vitamin E. Yes/No

h)for the same group of patients assess the effectiveness of a diet regimen by comparing the weights before and after. Yes/No

Fill in the blank:

(2) In a paired data set, the ___________between the two matched values can be used as a variable to run 1-sample t-test

(3) What is the null hypothesis, (Ho), and the alternate hypothesis, (H_A), for the paired data for a two-sided test for testing "no difference" or testing for a significant difference? (We will use _diff in place of , used in the textbook)

(4) What is the null hypothesis, (Ho), and the alternate hypothesis, (H_A), for the paired data for a one-sided test for testing the hypothesis for the values in the first set are lower than the values in the second set?

(5) What is the null hypothesis, (Ho), and the alternate hypothesis, (H_A), for the paired data for a one-sided test for testing the hypothesis for the values in the first set are higher than the values in the second set?

(6) Use the wetsuit and swimsuit velocities (m/sec) matched pairs data in the table below, to test whether there is a significant difference between the wet suit velocity and the swimsuit velocity.

Swimmer no.

Wetsuit Velocity

Swimsuit Velocity

1

1.57

1.49

2

1.47

1.37

3

1.42

1.35

4

1.35

1.27

5

1.22

1.12

6

1.75

1.64

7

1.64

1.59

8

1.57

1 52

9

1.56

1.50

10

1.53

1.45

11

1.49

1.44

12

1.51

1.41

(a)Give the null and the alternative hypotheses

(b) Use Minitab to run a paired t-test for the Minitab> Enter the wet suit velocity and the swim suit velocity data in columns 1 and 2 (with 12 data values in each column)> Basic Statistics>Paired t...>Each Sample is in a column> Sample 1>Double-click wetsuit> Sample 2>Double-click swim suit>Options (keep defaults)>Hypothesized difference = 0.0 >Alternative Hypothesis > Difference hypothesized diff. > OK (Copy your Minitab output below)

(c)What is the t-value?

(d)What is the p-value?

(e)Based on the p-value, can the null hypothesis be rejected at 5% level of significance?

(f)What is the conclusion about the mean difference between the swim velocities for the wet suit and the swim suit?

(7) Use the wetsuit and swimsuit velocities (m/sec) matched-pairs data to test whether the wetsuit velocities are significantly higher than swimsuit velocities.

(a)Give the null and the alternative hypotheses

(b) Use Minitab to run a paired t-test. Open Minitab> Enter the wet suit velocity and the swimsuit velocity data in columns 1 and 2 (with 12 data values in each column)> Basic Statistics>Paired t...>Each Sample is in a column> Sample 1>Double-click wet suit> Sample 2>Double-click swim suit>Options >Hypothesized difference = 0.0 >Alternative Hypothesis >Difference > hypothesized diff. > OK (Copy your Minitab output below)

(c)What is the t-value?

(d)What is the p-value?

(e)Based on the p-value, can the null hypothesis be rejected at 5% level of significance?

(f)What is the conclusion are the wetsuit velocities significantly higher than for the swimsuit velocities?

Hypothesis Testing when the mean and the standard deviation of differences between the matched pairs are known.

(8)A group of 62 people were used in a blood pressure study. The systolic blood pressure of each person was taken after they had been standing for five minutes. After one hour the systolic blood pressure for each person was taken again, this time after they had been reclining (supine) in bed for five minutes. The data is paired (matched) since the two sets of blood pressure readings were taken on the same individuals. The mean and the standard deviation of the sample differences are given below.Test the hypothesis that the mean systolic blood pressure is lower after reclining (supine) in bed for 5-minutes compared to the mean systolic blood pressure after standing for 5 minutes. (In other words, test the hypothesis that the difference between the two is (negative) or less than zero.)

Mean _{diff =} - 10.83Stand. Dev. = 29.54

(a)Give the null and the alternative hypotheses

(b) We will use 1-sample t-test in this case.) Minitab>Stat> Basic Statistics>1-sample t >Summarized Data> >Sample size > 62 > Sample mean > -10.83>Standard deviation 29.54>Select Perform Hypothesis Test> Hypothesized mean=0> Options >Alternative Hypothesis >Mean OK>OK

(c)What is the t-value?

(d)What is the p-value?

(e)Based on the p-value, can the null hypothesis be rejected at 5% level of significance?

(f)What is the conclusion regarding the investigation?

Section 5.3

Two Sample Test for Independent Data

_1: represents the population mean for the first population, and

_2: represents the population mean for the second population.

(9) What is the null hypothesis, (Ho), and the alternate hypothesis, (H_A), for the independent samples data for the two-sided test for testing "no difference" or significant difference between the two means?

(10) What is the null hypothesis, (Ho), and the alternative/research hypothesis for the one-sided test for testing the hypothesis to test whether the mean of the population 1 is higher than the mean of the population 2?

(₁ > ₂)

(11) What is the null hypothesis, (Ho), and the alternative/research hypothesis for the one-sided test for testing the hypothesis to test whether the mean of population 1, is lower than the mean of population 2 ? (₁ < ₂)?

Hypothesis Test for Independent Samples:

(12) Using the information in the table below conduct a hypothesis test to test whether there is an evidence that newborns from mothers who smoke have a different mean birth weight than newborns from mothers who do not smoke.

Smoking status of moms of newborns

Smoker

Non-smoker

mean

6.78

7.18

st. dev (SD)

1.43

1.60

Sample size

50

100

(a)Write the null and the alternative hypotheses

(b) Minitab>Stat>Basic Statistics>2-Sample t...>Summarized Data> For Sample 1 >Sample size >50> Sample mean >6.78 >Standard deviation >1.43 >For Sample 2 >Sample size >100> Sample mean >7.18>Standard deviation >1.60 >Options>Confidence level 95.0>Hypothesized difference: 0>Alternative Hypothesis> Difference hypothesized difference>OK>OK

(c)What is the t-value?

(d)What is the p-value?

(e)Based on the p-value, can the null hypothesis be rejected at 5% level of significance?

(f)What is the conclusion about the difference between mean birth weights of newborns whose mothers smoke compared to the mean birth weights of newborns whose mothers are non-smokers?

A couple of points to note:

If you need to conduct a hypothesis test when a Two Sample t-test with Minitab when the data is in two columnsuse these steps: Minitab:> Open Data Set >Stat> Basic Statistics>2-sample t>Each Sample is in its own column> Sample 1>Double-click ______> Sample 2>Double-click ________>Options (keep defaults)>Hypothesized difference = 0.0 >Alternative Hypothesis > Difference hypothesized diff. (or the appropriate hypothesis> OK>OK

For 2-sample test with Minitab, the difference between the two is calculated as:

Sample 1 - Sample2