Question

An experiment was conducted to test the effect of a new drug on a viral infection. After the infection was induced in 100 mice, the mice were randomly split into two groups of 50. The first group, the control group, received no treatment for the infection, and the second group received the drug. After a 30-day period, the proportions of survivors, p?1 and p?2, in the two groups were found to be 0.36 and 0.64, respectively.

(a) Is there sufficient evidence to indicate that the drug is effective in treating the viral infection? Use ? = 0.05.

State the null and alternative hypotheses.

H0: (p1 ? p2) < 0 versus Ha: (p1 ? p2) > 0

H0: (p1 ? p2) = 0 versus Ha: (p1 ? p2) < 0

H0: (p1 ? p2) ? 0 versus Ha: (p1 ? p2) = 0

H0: (p1 ? p2) = 0 versus Ha: (p1 ? p2) ? 0

H0: (p1 ? p2) = 0 versus Ha: (p1 ? p2) > 0

Find the test statistic and rejection region. (Round your answers to two decimal places. If the test is one-tailed, enter NONE for the unused region.)

test statistic: z =

rejection region: z >, z <

State your conclusion.

H0 is not rejected. There is insufficient evidence to indicate that the drug is effective in treating the viral infection.

H0 is rejected. There is sufficient evidence to indicate that the drug is effective in treating the viral infection.

H0 is not rejected. There is sufficient evidence to indicate that the drug is effective in treating the viral infection.

H0 is rejected. There is insufficient evidence to indicate that the drug is effective in treating the viral infection.

(b) Use a 95% confidence interval to estimate the actual difference (p1 ? p2) in the survival rates for the treated versus the control groups. (Round your answers to two decimal places.)

_____ to _____

You may need to use the appropriate appendix table or technology to answer this question.

Answer 1

Solution:-

a)

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: P₁> P₂
Alternative hypothesis: P₁ < P₂

Note that these hypotheses constitute a one-tailed test.

Formulate an analysis plan. For this analysis, the significance level is 0.05. The test method is a two-proportion z-test.

Analyze sample data. Using sample data, we calculate the pooled sample proportion (p) and the standard error (SE). Using those measures, we compute the z-score test statistic (z).

p = (p₁ * n₁ + p₂ * n₂) / (n₁ + n₂)

p = 0.50

SE = sqrt{ p * ( 1 - p ) * [ (1/n₁) + (1/n₂) ] }
SE = 0.10
z = (p₁ - p₂) / SE

z = - 2.80

where p₁ is the sample proportion in sample 1, where p₂ is the sample proportion in sample 2, n₁ is the size of sample 1, and n₂ is the size of sample 2.

Since we have a one-tailed test, the P-value is the probability that the z-score is less than -2.80.

Thus, the P-value = 0.003

Interpret results. Since the P-value (0.003) is less than the significance level (0.05), we have to reject the null hypothesis.

H0 is rejected. There is sufficient evidence to indicate that the drug is effective in treating the viral infection.

b) 95% confidence interval to estimate the actual difference (p1 ? p2) in the survival rates for the treated versus the control groups is C.I = (-0.476, - 0.084).

C.I = 0.36 - 0.64 + 1.96 × 0.10

C.I = - 0.28 + 0.196

C.I = (-0.476, - 0.084)