A statistician demonstrated how to test the efficacy of an HIV vaccine. He repor
ID: 3043252 • Letter: A
Question
A statistician demonstrated how to test the efficacy of an HIV vaccine. He reported the results in the 2 times ×2 table shown below. The trial consisted of 8 AIDS patients vaccinated with the new drug and 31 AIDS patients who were treated with a placebo (no vaccination). The table shows the number of patients who tested positive and negative for the MN strain in the trial follow-up period. Complete parts a through e.
Positive Negative totals
Unvaccinated 24 9 33
Vaccinated 2 6 8
Totals 26 15 41
Conduct a test to determine whether the vaccine is effective in treating the MN strain of HIV. Use
=0.05.
What are the null and alternative hypotheses?
Find the test statistic. 2=
Specify the rejection region. Choose the correct answer below.
A. 2greater than>3.84146
B. 2greater than>16.9190
C. 2greater than>9.48773
D. 2greater than>15.5073
Choose from the following:
A. Reject Upper H 0H0. There is insufficient evidence to indicate that the vaccine is effective in treating the MN strain of HIV at =0.05.
B. Reject Upper H 0H0. There is sufficient evidence to indicate that the vaccine is effective in treating the MN strain of HIV at =0.05.
C.Fail to reject Upper H 0H0. There is insufficient evidence to indicate that the vaccine is effective in treating the MN strain of HIV at =0.05.
Are the assumptions for the test you carried out in part a, satisfied?
Yes
No
. In the case of a 2 times ×2 contingency table, R. A. Fisher developed a procedure for computing the exact p-value for the test. The method utilizes the hypergeometric probability distribution. Consider the hypergeometric probability shown on the right which represents the probability that
2 out of 8 vaccinated AIDS patients test positive and
24 out of 33 unvaccinated patients test
Positive long dash—that
is, the probability of the result shown in table, given that the null hypothesis of independence is true.
Compute this probability.
StartFraction left parenthesis Start 2 By 1 Matrix 1st Row 1st Column 8 2nd Row 1st Column 2 EndMatrix right parenthesis left parenthesis Start 2 By 1 Matrix 1st Row 1st Column 33 2nd Row 1st Column 24 EndMatrix right parenthesis Over left parenthesis Start 2 By 1 Matrix 1st Row 1st Column 41 2nd Row 1st Column 26 EndMatrix right parenthesis EndFraction
(8
(2
(33
(24
(41
(26
The probability is p=
.(Round to four decimal places as needed.)
d. Refer to part c.
Two contingency tables that are more unsupportive of the null hypothesis of independence than the observed table are shown in the accompanying table. Explain why these tables provide more evidence to reject Upper H 0H0 than the original table does.
If vaccine and MN strain are independent, then the proportion of positive results should be
relatively the same
different
for both patient groups.
In the two tables presented, the proportion of positive results for the vaccinated group is
smaller
greater
than the proportion for the original table.
Compute the probability of the first table, using the hypergeometric formula.
The probability of the contingency table is
. (Round to four decimal places as needed.)
Compute the probability of the second table, using the hypergeometric formula.
The probability of the contingency table is
(Round to four decimal places as needed.)
e. The p-value of Fisher's exact test is the probability of observing a result at least as unsupportive of the null hypothesis as is the observed contingency table, given the same marginal totals. Sum the probabilities of parts c and d to obtain the p-value of Fisher's exact test.
The p-value of the test is.
(Round to four decimal places as needed.)
Interpret this value in the context of the vaccine trial. Choose the correct answer below.
A. Since the p-value is greater than alpha, there is insufficient evidence to indicate that the vaccine is effective in treating the MN strain of HIV at =0.05.
B. Since the p-value is less than alpha, there is insufficient evidence to indicate that the vaccine is effective in treating the MN strain of HIV at =0.05.
C. Since the p-value is less than alpha, there is sufficient evidence to indicate that the vaccine is effective in treating the MN strain of HIV at =0.05
. In the case of a 2 times ×2 contingency table, R. A. Fisher developed a procedure for computing the exact p-value for the test. The method utilizes the hypergeometric probability distribution. Consider the hypergeometric probability shown on the right which represents the probability that
2 out of 8 vaccinated AIDS patients test positive and
24 out of 33 unvaccinated patients test
Positive long dash—that
is, the probability of the result shown in table, given that the null hypothesis of independence is true.
Compute this probability.
StartFraction left parenthesis Start 2 By 1 Matrix 1st Row 1st Column 8 2nd Row 1st Column 2 EndMatrix right parenthesis left parenthesis Start 2 By 1 Matrix 1st Row 1st Column 33 2nd Row 1st Column 24 EndMatrix right parenthesis Over left parenthesis Start 2 By 1 Matrix 1st Row 1st Column 41 2nd Row 1st Column 26 EndMatrix right parenthesis EndFraction
(8
(2
(33
(24
(41
(26
The probability is p=
.(Round to four decimal places as needed.)
Explanation / Answer
Answer:
A statistician demonstrated how to test the efficacy of an HIV vaccine. He reported the results in the 2 times ×2 table shown below. The trial consisted of 8 AIDS patients vaccinated with the new drug and 31 AIDS patients who were treated with a placebo (no vaccination). The table shows the number of patients who tested positive and negative for the MN strain in the trial follow-up period. Complete parts a through e.
Positive Negative totals
Unvaccinated 24 9 33
Vaccinated 2 6 8
Totals 26 15 41
Chi-Square Test
Observed Frequencies
Column variable
Calculations
Row variable
C1
C2
Total
fo-fe
R1
24
9
33
3.0732
-3.0732
R2
2
6
8
-3.0732
3.0732
Total
26
15
41
Expected Frequencies
Column variable
Row variable
C1
C2
Total
(fo-fe)^2/fe
R1
20.9268
12.0732
33
0.4513
0.7823
R2
5.0732
2.9268
8
1.8616
3.2268
Total
26
15
41
Data
Level of Significance
0.05
Number of Rows
2
Number of Columns
2
Degrees of Freedom
1
Results
Critical Value
3.8415
Chi-Square Test Statistic
6.3220
p-Value
0.0119
Reject the null hypothesis
Conduct a test to determine whether the vaccine is effective in treating the MN strain of HIV. Use
=0.05.
What are the null and alternative hypotheses?
Ho: the vaccine is not effective in treating the MN strain of HIV
H1: the vaccine is effective in treating the MN strain of HIV
Find the test statistic. 2=6.3220
Specify the rejection region. Choose the correct answer below.
Answer: A. 2greater than>3.84146
B. 2greater than>16.9190
C. 2greater than>9.48773
D. 2greater than>15.5073
Choose from the following:
A. Reject Upper H 0H0. There is insufficient evidence to indicate that the vaccine is effective in treating the MN strain of HIV at =0.05.
Answer: B. Reject H0. There is sufficient evidence to indicate that the vaccine is effective in treating the MN strain of HIV at =0.05.
C.Fail to reject Upper H 0H0. There is insufficient evidence to indicate that the vaccine is effective in treating the MN strain of HIV at =0.05.
Are the assumptions for the test you carried out in part a, satisfied?
Yes
Answer: No
( one expected value less than 5).
Chi-Square Test
Observed Frequencies
Column variable
Calculations
Row variable
C1
C2
Total
fo-fe
R1
24
9
33
3.0732
-3.0732
R2
2
6
8
-3.0732
3.0732
Total
26
15
41
Expected Frequencies
Column variable
Row variable
C1
C2
Total
(fo-fe)^2/fe
R1
20.9268
12.0732
33
0.4513
0.7823
R2
5.0732
2.9268
8
1.8616
3.2268
Total
26
15
41
Data
Level of Significance
0.05
Number of Rows
2
Number of Columns
2
Degrees of Freedom
1
Results
Critical Value
3.8415
Chi-Square Test Statistic
6.3220
p-Value
0.0119
Reject the null hypothesis
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