57.8k views
2 votes
Construct the indicated confidence interval for the difference between population proportions Assume that the samples are independent and that they have been randomly selected. Be sure to provide the (1) Null & Alternative Hypotheses, (2) Test Statistic, (3) P-value (or critical value), and (4) concluding statements.

x1 = 15, n1 = 50 and x2 = 23, n2 = 60;

Construct a 90% confidence interval for the difference between population proportions p1 - p2.

User Alka
by
7.6k points

1 Answer

3 votes

(1) Null and Alternative Hypotheses:

The null hypothesis is that the difference between the population proportions is zero (i.e., p1 - p2 = 0). The alternative hypothesis is that the difference between the population proportions is not zero (i.e., p1 - p2 ≠ 0).

(2) Test Statistic:

We can use the z-test for the difference between two proportions to calculate the test statistic:

z = (p1 - p2) - 0 / sqrt(p(1-p)(1/n1 + 1/n2))

where p = (x1 + x2) / (n1 + n2)

Substituting the given values, we have:

p1 = 15/50 = 0.3, p2 = 23/60 = 0.383, n1 = 50, n2 = 60

p = (15 + 23) / (50 + 60) = 0.2917

z = (0.3 - 0.383) - 0 / sqrt(0.2917*(1-0.2917)*(1/50 + 1/60)) ≈ -1.61

(3) P-value:

The two-sided P-value for the test is the probability of observing a test statistic as extreme as -1.61, assuming the null hypothesis is true. From the z-table, the probability of observing a z-value less than -1.61 is approximately 0.053, so the probability of observing a test statistic as extreme as -1.61 (in either direction) is approximately 0.106.

(4) Conclusion:

To construct the 90% confidence interval, we need to find the margin of error and the interval endpoints:

Margin of error = z*(standard error) = 1.645*sqrt(p(1-p)(1/n1 + 1/n2)) ≈ 0.122

Lower endpoint: (p1 - p2) - margin of error = 0.3 - 0.383 - 0.122 ≈ -0.205

Upper endpoint: (p1 - p2) + margin of error = 0.3 - 0.383 + 0.122 ≈ -0.027

Therefore, the 90% confidence interval for the difference between population proportions is approximately (-0.205, -0.027).

Since the interval does not contain the value zero, we can conclude that we have evidence to reject the null hypothesis and accept the alternative hypothesis that the difference between the population proportions is not zero at the 0.10 level of significance.

User Rohan Rao
by
7.6k points