Section 9.5 Day 3

Key Question ?

Do we really have two independent samples Key Question Do we really have two independent samples or

Do we really have two independent samples Key Question Do we really have two independent samples or do we have only one sample of paired data?

Key Question Do we really have two independent samples (difference of two means) or do we have only one sample of paired data?

Use 2-sample t procedure Key Question Do we really have two independent samples (difference of two means) Use 2-sample t procedure or do we have only one sample of paired data?

Use 2-sample t procedure Key Question Do we really have two independent samples (difference of two means) Use 2-sample t procedure or do we have only one sample of paired data? (mean difference based on paired observations)

Use 2-sample t procedure Use 1-sample t procedure Key Question Do we really have two independent samples (difference of two means) Use 2-sample t procedure or do we have only one sample of paired data? (mean difference based on paired observations) Use 1-sample t procedure

Where do experiments fall? What is always true about an experiment?

Where do experiments fall? What is true about an experiment? Treatments must be randomly assigned.

Where do experiments fall? Do we really have two independent samples or do we have only one sample of paired data?

Where do experiments fall? Do we really have two independent samples Treatments randomly assigned to all available units/subjects or do we have only one sample of paired data?

Where do experiments fall? Do we really have two independent samples Treatments randomly assigned to all available units/subjects or do we have only one sample of paired data? Treatments randomly assigned within each matched pair

Where do experiments fall? Do we really have two independent samples Treatments randomly assigned to all available units/subjects or do we have only one sample of paired data? Treatments randomly assigned within each matched pair Each subject receives all treatments in random order (repeated measures)

Page 653, P34 (a)

Page 653, P34 (a) There is no reason to pair a particular bird with a particular fish, so these are independent samples.

Page 655, E70

Page 655, E70 a. Should you use a test of the difference of two means or a test of the mean of the differences?

Page 655, E70 You should use a one-sample test of the mean of the differences. We do not have two independent samples. Because the measurements of the distance are for the fifth launch and tenth launch for each team, this is a repeated measures design.

Page 655, E70 Should this be a one-sided test or a two-sided test?

Page 655, E70 a. A one-sided test should be used because you want to test whether the teams improved, which would mean the distance on the tenth launch would be longer.

Page 655, E70 b. Check conditions:

Page 655, E70 b. Check conditions: This is not a random sample of teams and the paired observations are not in random order. So, what is this?

Page 655, E70 b. Check conditions: This is not a random sample of teams and the paired observations are not in random order. This is an observational study of teams.

Page 655, E70 b. Check conditions: (Use STAT PLOT of L3) The boxplot of the differences is basically symmetric with no outliers so . . . . Be sure to show your plot.

Page 655, E70 b. Check conditions: The boxplot of the differences is basically symmetric with no outliers so it is reasonable to assume the differences came from a normally distributed population. Be sure to show your plot.

Page 655, E70 b. Check conditions: We do not know if there were 60 or more teams that launched gummy bears.

Page 655, E70 b. Check conditions: The conditions have not been met for a significance test for paired differences but the test result may still be useful.

Page 655, E70 State your hypotheses: H0 :

Page 655, E70 State your hypotheses: H0 : μd = 0, where μd is the mean of the differences between the distance on the tenth launch and the distance on the fifth launch

Statistically significant evidence that teams improved? Page 655, E70 State your hypotheses: H0 : μd = 0, where μd is the mean of the differences between the distance on the tenth launch and the distance on the fifth launch Ha : μd > 0 Statistically significant evidence that teams improved?

Page 655, E70 T-Test Inpt: Data μo: 0 List: L3 (10th – 5th) Freq: 1 Calculate

Page 655, E70 T-Test Inpt: Data μo: 0 List: L3 Freq: 1 μ: > μo Calculate t ≈ 1.477 and P-value ≈ 0.0998

Page 655, E70 Write your conclusion in context, linked to your computations: I do not reject the null hypothesis because the P-value of 0.0998 is greater than the significance level of = 0.01.

Page 655, E70 Write your conclusion in context, linked to your computations: I do not reject the null hypothesis because the P-value of 0.0998 is greater than the significance level of = 0.01. There is not sufficient evidence to support the claim that the teams improved from launch 5 to launch 10.

Page 655, E70 c. What type of error may have been made?

Page 655, E70 c. Type I: reject true null hypothesis Type II: do not reject false null hypothesis

Page 655, E70 c. A Type II error could have been made because if practice still has an effect after the 5th launch, you have failed to reject a false null hypothesis. Type I: reject true null hypothesis Type II: do not reject false null hypothesis

Page 655, E70 d. The result may not match your intuition. You would still expect some improvement with practice, although you would expect less improvement from the fifth launch to the tenth than from the first launch to the tenth. At some point you might expect that more practice is not going to cause any more noticeable improvement, but you might not expect that to occur already after the fifth launch.

Page 656, E74

Page 656, E74 a. Supposing the data are not paired, the treatments were assigned randomly to all the subjects.

Page 656, E74 a. The distributions of both lists are symmetric with no outliers so it’s reasonable to assume both samples came from normally distributed populations.

Page 656, E74 a. Experiment so no third condition. The conditions are met for a two-sample significance test. Name of test?

Page 656, E74 a. Experiment so no third condition. The conditions are met for a two-sample significance test. Two-sided significance test for difference of two means.

Page 656, E74 2-SampTTest Inpt: Data List 1: L1 List 2: L2 Freq1: 1 μ1: ≠ μ2 (does mean number of words differ?) Pooled: No

Page 656, E74 2-SampTTest Inpt: Data List 1: L1 List 2: L2 Freq1: 1 μ1: ≠ μ2 Pooled: No t ≈ ±1.38 and P-value ≈ 0.174

Page 656, E74

Page 656, E74 c. Now we have repeated measures design. Name of test?

Page 656, E74 c. Now we have repeated measures design. Name of test? Two-sided significance test of the mean difference based on paired observations

Page 656, E74 c. Now we have repeated measures design. Treatments assigned in random order to each subject.

Page 656, E74 c. Boxplot of differences is fairly symmetric with no outliers so reasonable to assume differences came from normally distributed population. (Use STAT PLOT of L3)

Page 656, E74 L3 = L1 – L2 Use T-Test Inpt: Data L3

Page 656, E74 t ≈ ± 2.17 P-value ≈ 0.04

Page 656, E74

Page 656, E74 e. The one-sample test on the differences has more power because the P-value is smaller, meaning the null hypothesis is more likely to be rejected. Difference of means: P-value ≈ 0.174 Mean difference: P-value ≈ 0.04

Page 636, E61 Should you use a two-sample test of the difference of two means or a one-sample test of the mean of the differences? Why?

Page 636, E61 Should you use a two-sample test of the difference of two means or a one-sample test of the mean of the differences? Why? These are random samples and presumably independent because there is no indication of any pairing taking place.

Questions?

These rates of women’s participation in the labor force were collected by the U.S. Department of Labor Statistics in 19 cities for 1968 and 1972. The purpose was to monitor the change in the percentage of women who were in the labor force during that period.

a. Discuss whether a statistical test is appropriate for determining whether there was a change in the mean percentage of women in the labor force, by city, between 1968 and 1972.

a. Discuss whether a statistical test is appropriate for determining whether there was a change in the mean percentage of women in the labor force, by city, between 1968 and 1972. The sample is not random. These are the largest U.S. cities. Thus, they are unlikely to be representative of all cities in the entire United States. Thus, a test of significance based on matched pairs isn’t really appropriate.

b. Regardless of your response to part a, carry out a test of whether there is statistically significant evidence of a change in the mean percentage of women who were in the labor force, by city, between 1968 and 1972.

Name Test and Check Conditions

Name Test and Check Conditions Two-sided significance test of a mean difference based on paired observations

Check Conditions The sample of cities wasn’t selected at random. These are the largest cities in the United States and there is no reason to suppose that these large cities are representative of all cities or of the entire United States.

Check Conditions The set of differences is slightly skewed with no outliers, but this doesn’t indicate a serious problem with non-normality.

Check Conditions The population is at least 10 times the sample size as there are more than 190 large cities in the U.S.

State Hypotheses H0: = 0, the mean difference in the percentage of women who participated in the labor force, by city, between 1968 and 1972, is 0.

State Hypotheses H0: = 0; the mean difference in the percentage of women who participated in the labor force, by city, between 1968 and 1972, is 0 Ha:

Do Computations

Do Computations

Conclusion in Context I reject the null hypothesis because the P-value of 0.02435 is less than the significance level of 0.05.

Conclusion in Context I reject the null hypothesis because the P-value of 0.02435 is less than the significance level of 0.05. There is sufficient evidence to support the claim that there was a significant change in the mean percentage of women who were in the labor force between 1968 and 1972 for these 19 cities.

Conclusion in Context There is sufficient evidence to support the claim that there was a significant change in the mean percentage of women who were in the labor force, by city, between 1968 and 1972 for these 19 cities. Note: we can not make an inference to all cities because we did not have a random sample of cities.

Strength of Evidence Fathom Lab 9.4a

Strength of Evidence Evidence of a difference is stronger when: 1) the difference in means is smaller or larger? 2) When the spreads are smaller or larger? 3) When the sample sizes are smaller or larger?

Strength of Evidence Evidence of a difference is stronger when: 1) the difference in means is smaller or larger? 2) When the spreads are smaller or larger? 3) When the sample sizes are smaller or larger?

Strength of Evidence Evidence of a difference is stronger when: 1) the difference in means is smaller or larger? 2) When the spreads are smaller or larger? 3) When the sample sizes are smaller or larger?

Strength of Evidence Evidence of a difference is stronger when: 1) the difference in means is smaller or larger? 2) When the spreads are smaller or larger? 3) When the sample sizes are smaller or larger?