1A. In August 2017, the Annenberg Public Policy Center asked the equivalent of a
ID: 3303967 • Letter: 1
Question
1A. In August 2017, the Annenberg Public Policy Center asked the equivalent of a simple random sample of 705 U.S. adults the following: “Would you mind naming any of (the three branches of government)?” The equivalent results are summarized below.
- 183 named all three branches
- 92 named two of the three branches
- 190 named one branch
- 233 could not name any branches
- 7 refused to answer
A political scientist believes that at least half of U.S. adults can name at least two branches of government. What is the (one-sided) p-value for testing that hypothesis?
A. 10–9
B. 10–37
C. 0.004
D. 0.486
E. 0.08
1B. What is the correct interpretation for the p-value?
A. Since p > 0.05, fail to reject H0, and conclude that at least half of U.S. adults can name at least two branches of government.
B. Since p < 0.05, reject H0, and conclude that at least half of U.S. adults can name at least two branches of government.
C. Since p < 0.05, reject H0, and conclude that less than half of U.S. adults can name at least two branches of government.
D. Since p > 0.05, reject H0, and conclude that the percentage of U.S. adults who can name at least two branches of government might be greater than 50% or might be less than 50%
E. Since p < 0.05, fail to reject H0, and conclude that the percentage of U.S. adults who can name at least two branches of government must not be exactly equal to 50%.
A. 10–9
B. 10–37
C. 0.004
D. 0.486
E. 0.08
Explanation / Answer
Solution:-
x = 183 + 92 = 275
x = 275
n = 705
p = 275/705
p = 0.390
State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.
Null hypothesis: P > 0.50
Alternative hypothesis: P < 0.50
Note that these hypotheses constitute a one-tailed test. The null hypothesis will be rejected only if the sample proportion is too small.
Formulate an analysis plan. For this analysis, the significance level is 0.05. The test method, shown in the next section, is a one-sample z-test.
Analyze sample data. Using sample data, we calculate the standard deviation () and compute the z-score test statistic (z).
= sqrt[ P * ( 1 - P ) / n ]
= 0.0184
z = (p - P) /
z = - 5.98
where P is the hypothesized value of population proportion in the null hypothesis, p is the sample proportion, and n is the sample size.
Since we have a one-tailed test, the P-value is the probability that the z-score is less than - 5.98. We use the Normal Distribution Calculator to find P(z < - 5.98) = 0.00003
Interpret results. Since the P-value (almost 0) is less than the significance level (0.05), we cannot accept the null hypothesis.
C. Since p < 0.05, reject H0, and conclude that less than half of U.S. adults can name at least two branches of government.
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