16. A local school board member randomly sampled private and public high school
ID: 3316452 • Letter: 1
Question
16. A local school board member randomly sampled private and public high school teachers in his district to compare the proportions of National Board Certified teachers in the faculty. The results were: Private Schools Public Schools sample size: proportion of NBC teachers: n1 = 80 h-0.175 n2 = 520 -0.150 (a) State the null and alternative hypotheses to test whether the proportion of NBC teachers in private schools exceeds that of public schools (b) State the formula for the test statistic and compute its value. Justify your answer (d) State a conclusion about the difference in proportions based on the test you performed. (e) Compute a p-value for this test.Explanation / Answer
Solution:-
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.
Formulate an analysis plan. For this analysis, the significance level is 0.10. 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 = 0.15333
SE = sqrt{ p * ( 1 - p ) * [ (1/n1) + (1/n2) ] }
SE = 0.0433
z = (p1 - p2) / SE
z = 0.58
zcritical = 1.96
Rejection region z > 1.96
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 two-tailed test, the P-value is the probability that the z-score is greater than 0.58.
Thus, the P-value = 0.281
Interpret results. Since the P-value (0.281) is greater than the significance level (0.05), we cannot accept the null hypothesis.
From the above test we can conclude that proportion of NBC teachers in private schools exceeds that of public schools.
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