AP Stats Sem 2 3.3.7 - 9, 3.4.3

Which of the following statements about pooling are true? 1. In statistics, to pool is to create an estimate of a common variance for two samples. Pooling gives narrower confidence intervals and lower P-values, so it yields a more precise final result with a higher power of the test. 2. Pooling can only be used in the rare instances when you can assume the populations are of equal size. 3. Without requiring that strong and often unrealistic conditions be met, computer software (or a calculator) will give a final result without pooling that's nearly as precise as the result you'd get if you pooled. 4. You can use pooling when the sample standard deviations are the same.

1 and 3

Say you're going to conduct a t procedure for the difference between the means of two populations. You decide the samples, of sizes 12 and 15, are independent. Which of the following statements are true? 1. If we don't pool variances, the number of degrees of freedom is 25. 2. If we don't pool variances, and we don't have access to software to compute degrees of freedom, the number of degrees of freedom is 11. 3. For a given level of confidence C, a confidence interval based on 11 degrees of freedom will be wider than the one based on 25 degrees of freedom.

1,2,and 3

Which of the following best describes the condition necessary to justify using a pooled estimator of the population variance?

Both populations have the same standard deviation

Researchers calculated the following summary statistics from simple random samples from two distinct populations: x1=3.647, s1=1.9, n1=17 x2=2.529, s2=1.7, n2=17 When the researchers computed the degrees of freedom by hand, they got k=16, and for the P-value, they got p=.0447. However, their computer software gave them k=31.60 and p=.0402. Which of the following is most likely true about the difference in p-values

Computer software gives a larger value for k and a lower p-value, which gives you a higher power of the test

Researchers randomly assign subjects to two groups, with each group receiving a different asthma medication. For one month, researchers record the number of asthma attacks suffered by each subject and compute the summary statistics as follows: x1= 3.8, s1 = 1.7, n1 = 23 x2 = 4.2, s2 = 1.5, n2 = 24 Which pair of hypotheses test whether the first mean is less than the second?

H0 : (µ1 - µ2) = 0 Ha : (µ1 - µ2) < 0

You take two random samples from two distinct populations and calculate the following statistics: x1=52.6, s1=3.2, n1=28 x2=29.3, s2=2.9, n2=29 k=degrees of freedom=27 t=4.07 p≈0.0002 State a conclusion, based on the p-value and a=.05, for the hypothesis test Ho: µ1 = µ2, Ha: µ1 > µ2.

Reject the null hypothesis in favor of the alternative, which states that the mean of population one is greater than the mean of population two

A backpack manufacturer wants to know if students in high school carry more books than college students do. Company researchers take a simple random sample from each group and record the number of textbooks each subject is carrying. They compute the following summary statistics: x̅1 = 3.647, s1 = 1.9, n1 = 17 x̅2 = 2.529, s2 = 1.7, n = 17 Using software, the researchers get p=.0402 for the hypothesis test Ho: µ1 = µ2, Ha: µ1 > µ2. Draw a conclusion for this hypothesis test using a=.05

We have significant evidence to reject the null in favor of the alternative hypothesis that high-school students carry more books

When you're estimating population means, or when you're doing hypothesis tests about population means, when is it appropriate to use the t procedures

When you don't know your population standard deviation and your sample shows no signs of extreme skewness or outliers.

Which of the following is true of the degrees of freedom, k, if you find them by taking the smaller of n1-1 when n1≠n2

When you use this estimate for k, you're less likely to reject a false null hypothesis

A backpack manufacturer wants to know if students in high school carry more books than college students do. Company researchers take a simple random sample from each group and record the number of textbooks each subject is carrying. They get the following data: High school: (5, 3, 2, 5, 6, 4, 7, 6, 5, 4, 3, 2, 1, 4, 3, 0, 2) College: (5, 3, 2, 4, 1, 0, 0, 3, 6, 2, 1, 3, 1, 2, 4, 4, 2) Using high-school students as sample one and college students as sample two, the researchers compute the following sample statistics and t statistic: x̅1 = 3.647, s1 = 1.9, n1 = 17 x̅2 = 2.529, s2 = 1.7, n = 17 t = 1.808 Calculate the degrees of freedon, k, using the conservative method, and use tcdf on your calculator to find the p-value for the hypothesis test Ho: µ1 = µ2, Ha: µ1 > µ2.

k=16, p=.0447

A backpack manufacturer wants to know if students in high school carry more books than college students do. Company researchers take a simple random sample from each group and record the number of textbooks each subject is carrying. They get the following data: High school: (5, 3, 2, 5, 6, 4, 7, 6, 5, 4, 3, 2, 1, 4, 3, 0, 2) College: (5, 3, 2, 4, 1, 0, 0, 3, 6, 2, 1, 3, 1, 2, 4, 4, 2) Using high-school students as sample 1 and college students as sample 2, the researchers compute the following sample statistics and t statistic: x̅1 = 3.647, s1 = 1.9, n1 = 17 x̅2 = 2.529, s2 = 1.7, n = 17 t = 1.808 Calculate the degrees of freedom and P-value for the hypothesis test Ho: µ1 = µ2, Ha: µ1 > µ2.

k=31.60, p=.0402

A research firm wants to determine whether there's a difference in married couples between what the husband earns and what the wife earns. The firm takes a random sample of married couples and measures the annual salary of each husband and wife. What procedure should the firm use to analyze the data for the mean difference in salary within married couples

one sample t procedure

What's the correct t statistic for the difference between means for a one-sample matched pairs procedure

t = (xbar n1 - n2 - µ₀) / (s/√n)

Researchers randomly assign subjects to two groups, with each group receiving a different asthma medication. For one month, researchers record the number of asthma attacks suffered by each subject and compute the summary statistics as follows: x1= 3.8, s1=1.7, n1=23 x2=4.2, s2=1.5, n2=24 Compute a t statistic and P-value for the hypothesis test H0:(m1-m2)=0, Ha :(m1-m2)<0. Use the conservative method to calculate your estimate for degrees of freedom, and don't pool or use your calculator. What did you get for your t statistic and your P-value, and what's your conclusion based on a=.05?

t= -.854 ; p=.2012 ; do not reject the null hypothesis that there's no difference between the two means.

Researchers randomly assign subjects to two groups, with each group receiving a different asthma medication. For one month, researchers record the number of asthma attacks suffered by each subject. Which of these is the correct procedure for measuring the difference in the number of asthma attacks between the two groups?

two-sample t procedure

A research firm wants to determine whether there's a difference in married couples between what the husband earns and what the wife earns. The firm takes a random sample of married couples and measures the annual salary of each husband and wife. What formula should the researchers use to find the confidence interval for the difference in means?

xbar sub n1-n2 ± t* s/√n

You draw two random samples from two distinct populations and calculate the following sample statistics: x1=52.6, s1=3.2, n1=28 x2=29.3, s2=2.9, n2=29 k=degrees of freedom=27 t=4.07 Using tcdf on your calculator, compute a p-value for the hypothesis test H0: m1=m2, Ha: m1>m2

.0002

A school district claims that the average teacher in the district earns $45,000 per year. The teacher's union disputes this claim and argues that the average salary is actually less. A random sample of 20 teachers yields a mean salary of $44,500 with a sample standard deviation of $1,750. What's the p-value for a test of the hypothesis that Ho: m=45,000 and Ha: m<45,000

.10<p<.15

Consider the following set of sample data: (34,32,34,32,40,37,31,31,29,27) We're interested in using this data to test a null hypothesis about the population mean. Which of the following statements are true? 1. Assuming this represents a random sample from the population, the sample mean is an unbiased estimator of the population mean. 2. We'd use t-procedures here if the population does not contain outliers or extreme skewness. 3. We'd use z-procedures here, since we're interested in the population mean

1 and 2

What's the critical value of t necessary to construct a 90% confidence interval for the difference between the means of two distinct populations of sizes 7 and 8? (Assume that the conditions necessary to justify pooling variances have been met)

1.771

A backpack manufacturer wants to know if students in high school carry more books than college students do. Company researchers take a simple random sample from each group and record the number of textbooks each subject is carrying. They get the following data: High school: (5, 3, 2, 5, 6, 4, 7, 6, 5, 4, 3, 2, 1, 4, 3, 0, 2) College: (5, 3, 2, 4, 1, 0, 0, 3, 6, 2, 1, 3, 1, 2, 4, 4, 2) Using high-school students as sample one and college students as sample two, what's you test statistic for the hypothesis test Ho: µ1 = µ2, Ha: µ1 > µ2.

1.808

A one-sided test of a hypothesis, based on a sample of size 9, yields a p-value of .035. Which of the following best describes the possible range of t values that yields this p-value

1.86 < t < 2.31

A two-sided test of a hypothesis, based on a sample of size 9, yields a p-value of .035. Which of the following best describes the possible range of t values that yields this p-value

2.45<t<2.90

What's the critical value of t(t*) needed to construct a 98% confidence interval for the mean of a distribution based on a sample of size 22

2.518

You draw two random samples from two distinct populations and calculate the following sample statistics: x1=52.6, s1=3.2, n1=28 x2=29.3, s2=2.9, n2=29 Using the conservative method, what are the degrees of freedom for a test of the difference between the two population means

27

Consider the following set of sample data: (34,32,34,32,40,37,31,31,29,27). Which of the following will give a 95% t confidence interval for the mean of the population from which the sample was drawn

32.7 +- 2.262(1.19)

A researcher draws two random samples from two distinct populations and calculates the following sample statistics: x̅1 = 52.6, s1 = 3.2, n1 = 28 x̅2 = 49.3, s2 = 2.9, n2 = 29 Compute a test statistic for the hypothesis test Ho: µ1 = µ2, Ha: µ1 > µ2

4.07

When estimating a population mean, you must use t procedures rather than normal curve procedures when:

The population standard deviation is unknown

We say that the t procedures are robust because:

They can be used even if some of the assumptions for their use are violated

You conduct a two-sided hypothesis test for the difference between two means based on a matched pairs comparison between two data sets of size 15, and get a t value of 2.55. Which of the following is true for this test?

This finding is significant at the .05 level but not at the .01 level