MATH 453 Essentials of Statistics Activity 1 Sample Z test and 1 Sample t test Name Purpose to illustrate what a p value is and to practice with hypothesis tests Recall that a p value is the probability that the statistic value difference could be as large or larger than that observed, and that the alternative hypothesis determines which direction matters Note Before completing this activity you may need to watch the screencasts StatCrunchZtest swf https youtu be gwn6bugPXRs StatCrunchTtest swf https youtu be yozGxeHjZk Problem Suppose we know the average number of cavities by age 30 years old for the US population is Approximately Normal Mean 7 Standard deviation 3 Residents of Happytown have been adding fluoride to their water for decades and would like to know whether this has lessened the incidence of cavities The parameter of interest is average number of cavities for all Happytown residents at age 30 Consider the sample mean of an SRS of n 9 Happytown residents who are 30 years old What are the null and alternative hypotheses for this test Assuming the null is true, what are the mean and standard deviation of the sample mean of an SRS of n 9 Happytown residents who are 30 years old, assuming that 3 for the Happytown residents Recall that for the sample mean The mean of the population mean jQuery22404678241983018583 1596725633103 The standard deviation of Consider the outcomes of two separate SRSs of size n 9 of Happytown residents, with cavities andcavities Use the mean and standard deviation of the sample mean from 2 above and StatCrunch ('Stat Calculators Normal') to produce Normal distribution plots (paste them here) depicting the probability of being less than or equal to (area to the left of) the sample mean for each sample does either sample seem to be evidence against the null Reading from the plot, what is the probability that the statistic value difference could be as great as that observed The probabilities in the Normal distribution plots are the p values for each test make a decision for each using 0 05 We should use a 1 sample Z test to conduct this hypothesis test Why Are the assumptions behind the test satisfied Now use StatCrunch to run the 2 Z tests, one Z test for each sample mean Paste the results here What do you conclude Do the p values agree with what you found in 4 Use StatCrunch to calculate the 95 confidence intervals corresponding to each test run in 6 Do the confidence intervals agree with the results of the tests in 6 How so Interpret the confidence intervals to elaborate upon the results of the tests Now suppose we no longer know the standard deviation of the number of cavities for all Happytown residents at age 30 but we do know the sample standard deviation S , so that we now know scores are Approximately Normal Mean 7 Sample standard deviation S 3 The residents of Happytown would still like to know whether adding fluoride to the drinking water has lessened the incidence of cavities, so that the hypotheses are the same We rely on the same two separate SRSs of size n 9 of Happytown residents, with cavities andcavities 8 We should use a 1 sample t test to conduct this hypothesis test Why Are the assumptions behind the test satisfied Use StatCrunch to run the two T tests, one T test for each sample mean Paste the results here What do you conclude Do the p values agree with what you found in 6 (they should be close but not exactly the same) Use StatCrunch to calculate the 95 confidence intervals corresponding to each test run in 9 Do the confidence intervals agree with the results of the tests in 9 How so Interpret the confidence intervals to elaborate upon the results of the tests How do the 95 confidence intervals corresponding to the t tests in 10 compare with the 95 confidence bounds corresponding to the Z tests in 7 Again, they should be similar but not exactly the same

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MATH 453 Essentials of Statistics Activity: 1 Sample Z -test and 1 Sample t -test Name: Purpose: to illustrate what a p-value is and to

MATH 453 Essentials of Statistics Activity:

1 Sample Z-test and 1 Sample t-test

Name:

Purpose: to illustrate what a p-value is and to practice with hypothesis tests. Recall that a p-value is the probability that the statistic - value difference could be as large or larger than that observed, and that the alternative hypothesis determines which direction matters.

Note: Before completing this activity you may need to watch the screencasts

StatCrunchZtest.swf https://youtu.be/gwn6bugPXRs

StatCrunchTtest.swf https://youtu.be/-yozGxeHjZk

Problem: Suppose we know the average number of cavities by age 30 years old for the US population is

-Approximately Normal

-Mean = 7

-Standard deviation = 3

Residents of Happytown have been adding fluoride to their water for decades and would like to know whether this has lessened the incidence of cavities. The parameter of interest is = average number of cavities for all Happytown residents at age 30. Consider the sample mean of an SRS of n = 9 Happytown residents who are 30 years old.

What are the null and alternative hypotheses for this test?

Assuming the null is true, what are the mean and standard deviation of the sample mean of an SRS of n = 9 Happytown residents who are 30 years old, assuming that = 3 for the Happytown residents? Recall that for the sample mean

-The mean of = the population mean = jQuery22404678241983018583_1596725633103??

-The standard deviation of = = ????

Consider the outcomes of two separate SRSs of size n = 9 of Happytown residents, with cavities andcavities. Use the mean and standard deviation of the sample mean from #2 above and StatCrunch ('Stat >> Calculators >> Normal') to produce Normal distribution plots (paste them here) depicting the probability of being less than or equal to (area to the left of) the sample mean for each sample - does either sample seem to be evidence against the null? Reading from the plot, what is the probability that the statistic - value difference could be as great as that observed?

The probabilities in the Normal distribution plots are the p-values for each test - make a decision for each using = 0.05.

We should use a 1-sample Z-test to conduct this hypothesis test. Why? Are the assumptions behind the test satisfied?

Now use StatCrunch to run the 2 Z-tests, one Z-test for each sample mean. Paste the results here. What do you conclude? Do the p-values agree with what you found in #4?

Use StatCrunch to calculate the 95% confidence intervals corresponding to each test run in #6. Do the confidence intervals agree with the results of the tests in #6? How so? Interpret the confidence intervals to elaborate upon the results of the tests.

Now suppose we no longer know the standard deviation of the number of cavities for all Happytown residents at age 30 but we do know the sample standard deviation S, so that we now know scores are

-Approximately Normal

-Mean = 7

-Sample standard deviation S = 3

The residents of Happytown would still like to know whether adding fluoride to the drinking water has lessened the incidence of cavities, so that the hypotheses are the same. We rely on the same two separate SRSs of size n = 9 of Happytown residents, with cavities andcavities.

8.We should use a 1-sample t-test to conduct this hypothesis test. Why? Are the assumptions behind the test satisfied?

Use StatCrunch to run the two T-tests, one T-test for each sample mean. Paste the results here. What do you conclude? Do the p-values agree with what you found in #6 (they should be close but not exactly the same)?

Use StatCrunch to calculate the 95% confidence intervals corresponding to each test run in #9. Do the confidence intervals agree with the results of the tests in #9? How so? Interpret the confidence intervals to elaborate upon the results of the tests.

How do the 95% confidence intervals corresponding to the t-tests in #10 compare with the 95% confidence bounds corresponding to the Z-tests in #7? Again, they should be similar but not exactly the same.