For each of the following three cases, explain (i) the hypotheses with a plausib
ID: 3310855 • Letter: F
Question
For each of the following three cases, explain (i) the hypotheses with a plausible definition of p1 and p2, (ii) whether or not the data indicate practical significance (use common sense and/or general knowledge), and (iii) whether or not the data indicate statistical significance. (a) A recent study of perfect pitch tested 2,700 students in American music conservatories. It found that 7% of non-Asian students and 32% of Asian students have perfect pitch. A two-sample Z-test of the difference in proportions resulted in a p-value of < 0.0001. (b) In July 1974, the PEW Research Center selected a large sample of voters. Sixty-six percent of those interviewed disapproved of President Nixon. In July 2007, the same PEW Research Center selected a comparable sample. In this case, 64.5% of those interviewed expressed disapproval of President Bush. The researchers pointed out that the p-value for comparing the two sample results was 0.023. (c) In a survey conducted in a statistics class at Boston College, students were asked their views on a number of social issues; 56% of the male students and 38% of the female students supported the death penalty. In a statistics lab, the students computed the corresponding p-value as 0.21.
Explanation / Answer
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 two-tailed test. The null hypothesis will be rejected if the proportion from population 1 is too big or if it is too small.
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).
It seems to be significance difference between two proportions.
Thus, the P-value = less than 0.0001
Interpret results. Since the P-value (almost 0) is less than the significance level (0.05), we cannot accept the null hypothesis.
Hence there is statistically significant difference between two proportions.
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. The null hypothesis will be rejected if the proportion from population 1 is too big or if it is too small.
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).
It seems to be no significance difference between two proportions.
Thus, the P-value = 0.023
Interpret results. Since the P-value (0.023) is less than the significance level (0.05), we cannot accept the null hypothesis.
Hence there is statistically significant difference between two proportions.
c)
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. The null hypothesis will be rejected if the proportion from population 1 is too big or if it is too small.
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).
It seems to be significance difference between two proportions.
Thus, the P-value = 0.21
Interpret results. Since the P-value (0.21) is more than the significance level (0.05), we have to accept the null hypothesis.
Hence there is no statistically significant difference between two proportions.
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