Question

3. According to the National Institute of Allergy and Infectious Diseases, approximately p = 0.08 of Americans suffer from hay fever. A random sample of individuals suffering from hay fever was obtained out an approximate population of N = 25,500,000, and each was treated with either a conventional antihistamine, ?1 = 255 or butterbur extract, ?2 = 237. The number of individuals who experienced relief from the conventional antihistamine was ?1 = 55 or ?̂1= 0.278 while the number of individuals who experienced relief from the butterbur extract was ?2 = 71 or ?̂2 = 0.232. Conduct the appropriate hypothesis test to determine if there is sufficient evidence to conclude that the proportion of individuals who experienced relief from the conventional antihistamine differed than the proportion of individuals who experienced relief from the butterbur extract. Use α = 0.01.
a. Step 1: Verify the assumptions of the Distribution of the Difference between Two Independent Sample Proportions, ?̂1- ?̂2 (3 pts):
• Both samples are randomly selected, or obtained through a randomized experiment
• Both samples are normally distributed, if:
o ?1?̂1(1- ?̂1)≥10:
o ?2?̂2(1- ?̂2)≥10:
• Samples are independent, if:
o ?1≤ 0.05 of N
o ?2≤ 0.05 of N
b. Step 2. Determine the hypotheses (1 pt.):
c. Step 3: Determine the level of significance, α (1pt.):
d. Step 4a: Determine ?̂ and calculate the test statistic (2 pts):
?̂= ?1+?2?1+?2
?0 = ?̂1- ?̂2 √?̂(1−?̂)√1?1+1?2
e. Step 4b. Determine the p-value associated with the test statistic (1 pt.):
f. Step 5: Compare the p-value to the alpha level α for the hypothesis test (1 pt.):\
g. Step 6: State the conclusion in a complete sentence (1 pt.):

Answer 1

Solution:-

a)
Both samples are randomly selected, or obtained through a randomized experiment
Both samples are normally distributed, since
o ?1?̂1(1- ?̂1)≥10
o ?2?̂2(1- ?̂2)≥10
Samples are independent, since
o ?1≤ 0.05 of N
o ?2≤ 0.05 of N

b)

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 two-tailed test.

c)

Formulate an analysis plan. For this analysis, the significance level is 0.01. 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).

d)

p = (p₁ * n₁ + p₂ * n₂) / (n₁ + n₂)

p = 0.2561

SE = sqrt{ p * ( 1 - p ) * [ (1/n₁) + (1/n₂) ] }
SE = 0.03938
z = (p₁ - p₂) / SE

z = - 2.13

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.

e)

Since we have a two-tailed test, the P-value is the probability that the z-score is less than -2.13 or greater than 2.13.

Thus, the P-value = 0.034

f) Interpret results. Since the P-value (0.034) is greater than the significance level (0.01), we cannot reject the null hypothesis.

g)

There is not sufficient evidence to conclude that that the proportion of individuals who experienced relief from the conventional antihistamine differed than the proportion of individuals who experienced relief from the butterbur extract.