Statistical Hypothesis Testing and Confidence Intervals

Posted on Aug 3, 2024 in Mathematics

1. Consider the three statements below. Draw a circle around any (if any) and all that are valid statistical hypotheses.

          H_A: p

_____    3.     Definition: the probability, assuming the null hypothesis is true, that a test statistic takes a value as extreme or more extreme than the value actually observed is which of the following?

               (D) P-value   

4.       In the article “Shooting Left the Right Way” it is reported that 63% of all skaters in the National Hockey League (NHL) are left-handed. If a simple random sample of 40 NHL skaters is selected and the proportion  who are left-handed determined, describe completely the sampling distribution of .

We have a simple random sample, and since  = 40(.63) = 25.2 and  = 40(1 – .63) = 14.8 are both greater than 10, the sample size is large enough for the CLT to apply.  So  = .63;  = .0763; and since the CLT applies, the shape is normal. Hence  ~ N(.63, .0763). 

5.       In the article “Shooting Left the Right Way” it is reported that 63% of all skaters in the National Hockey League (NHL) are left-handed. Looking at the list of the league’s top ten scorers, six are right-handed, so it is of interest to test to determine if the proportion of all NHL skaters who are left-handed is less than .63. State the appropriate null and alternative hypotheses that should be tested.

H₀:  = .63  versus  H_A:

6.       To test the hypotheses stated in question 5, a simple random sample of 25 NHL skaters was selected. Of these 25 skaters, 15 were left-handed. If appropriate, use this information to test the hypotheses stated in question 5 at the

          a = .10 level of significance. 

The assumptions are not met – = 25(1 – .63) = 9.25, which is not ≥ 10, so we can not complete the test.

7.       Of interest is p = the proportion of all Delta Sigma Theta members with a cumulative grade point average (GPA) of 3.5 or higher after the fall 2010 semester. An unsubstantiated claim is that the proportion of Delta Sigma Theta members with a cumulative GPA of 3.5 or higher after the fall 2010 semester is .40, and of interest is to test this versus the alternative that the proportion of all Delta Sigma Theta members with a cumulative GPA of 3.5 or higher after the fall 2010 semester is greater than .40. State the appropriate null and alternative hypotheses that should be tested.

H₀:  = .40  versus  H_A:  > .40

8.       Consider the information and hypotheses specified in question 7. A simple random sample of 32 Delta Sigma Theta members was selected, with 15 of these students having a cumulative GPA of 3.5 or higher. If appropriate, use this information to test the hypotheses stated in question 7 at the a = .05 level of significance.

 = .05 

We have a simple random sample, and both  = 32(.40) = 12.8 and  = 32(1 – .40) = 19.2 are > 10.

Z =  = 0.80

• p-value = P(Z

  0.80) = 1 – P(Z

(Calculator gives p-value = .2136)

 = .05   Since p-value > .05 we fail to reject H₀

• There is insufficient evidence that the proportion of all Delta Sigma Theta members with a cumulative GPA of 3.5 or higher after the fall 2010 semester is greater than .40
• 9.       Cynthia M. A. Butler-McIntyre is the 24^th National President of Delta Sigma Theta Sorority, Inc. Of interest is to estimate p = the proportion of all Delta Sigma Theta members who have met President Butler-McIntyre. To estimate this proportion, a 99% confidence interval will be calculated and the goal is that the margin of error will be no larger than .08. What is the minimum number of members that would need to be selected to allow the calculation of a 99% confidence interval with margin of error no greater than .08? It can be assumed for this problem only that the proportion of all Delta Sigma Theta members who met the 23^rd National President was .18 and this number can be used for this problem. Please circle your final answer

.18(1 – .18) = 153.04  round up to 154

                We would need a sample size of at least 154 members.

10.    A simple random sample of 200 Delta Sigma Theta members was selected and whether or not they had met President Butler-McIntyre was recorded for each. 26 out of these 200 Delta Sigma Theta members have met President Butler-McIntyre. If appropriate, use this information to calculate and interpret a 99% confidence interval for the proportion of all Delta Sigma Theta members who have met President Butler-McIntyre.

n = 200,  = .13, Z* for 99% confidence = 2.576

We have a simple random sample and both  = 200(.13) = 26 and  = 200(1 – .13) = 174 are greater than 10.  So the 99% confidence interval is

= (.0687, .1913)

          We have 99% confidence that the proportion of all Delta Sigma Theta members who have met President Butler-McIntyre is between .0687 and .1913.

_____ 11.  Is the proportion of all Delta Sigma Theta members who have met President Butler-McIntyre in the confidence interval computed in question 10?

• The proportion of all Delta Sigma Theta members who have met President Butler-McIntyre is not known and hence we do not know if it is in the interval or not
• _____    12.  In question 10 a confidence interval was computed based on a sample of 200 members. If the number of members in the sample were decreased to 160, what impact would this have on the margin of error and width of the confidence interval?

(D) Both the margin of error and the width would increase.