The null and alternate hypotheses are:
H0: π1 ≤ π2
H1: π1> π2
A sample of 100 observations from the first population indicated that X1 is 70. A sample of 150 observations from the second population revealed X2 to be 90. Use the .05 significance level to test the hypothesis.
(a) State the decision rule. (Round your answer to 2 decimal places.)
H0 is rejected if z >
(b) Compute the pooled proportion. (Round your answer to 2 decimal places.)
Pooled proportion
(c) Compute the value of the test statistic. (Round your answer to 2 decimal places.)
Value of the test statistic
To answer this question, we need to perform a hypothesis test for comparing two proportions and determine the decision rule, compute the pooled proportion, and compute the value of the test statistic.
(a) The decision rule can be determined by finding the critical value associated with the significance level of 0.05. Since the alternative hypothesis is one-sided (π1 > π2), it is a right-tailed test. We can use a Z-test, and the critical value can be found using a standard normal distribution table or a statistical software.
In this case, since H0 is π1 ≤ π2, the critical value corresponds to the z-score of the significance level (0.05) from the right tail.
The critical value is given by: z = invNorm(1 - α), where α is the significance level.
Using the standard normal distribution table or a statistical software, we can find the critical value to be approximately 1.645 (rounded to 2 decimal places).
Therefore, the decision rule can be stated as:
H0 is rejected if the test statistic (z-value) is greater than 1.645.
(b) The pooled proportion is calculated by combining the sample proportions from the two populations.
Pooled proportion (p) = (X1 + X2) / (n1 + n2)
where X1 is the number of successes in the first population, X2 is the number of successes in the second population, n1 is the sample size from the first population, and n2 is the sample size from the second population.
In this case:
X1 = 70 (from the first population)
X2 = 90 (from the second population)
n1 = 100 (sample size from the first population)
n2 = 150 (sample size from the second population)
Pooled proportion (p) = (70 + 90) / (100 + 150) = 160 / 250 = 0.64 (rounded to 2 decimal places).
Therefore, the pooled proportion is 0.64.
(c) The value of the test statistic for comparing two proportions is calculated using the following formula:
Test statistic = (p1 - p2) / sqrt(p*(1-p)*((1/n1) + (1/n2)))
where p1 and p2 are the sample proportions from the two populations, p is the pooled proportion, and n1 and n2 are the sample sizes from the two populations.
In this case:
p1 = X1 / n1 = 70 / 100 = 0.7 (proportion from the first population)
p2 = X2 / n2 = 90 / 150 = 0.6 (proportion from the second population)
p = 0.64 (pooled proportion)
n1 = 100 (sample size from the first population)
n2 = 150 (sample size from the second population)
Test statistic = (0.7 - 0.6) / sqrt(0.64 * (1-0.64) * ((1/100) + (1/150)))
Test statistic = 0.1 / sqrt(0.2304 * (0.3456) * (0.01 + 0.00667))
Test statistic = 0.1 / sqrt(0.0797952)
Test statistic ≈ 0.1 / 0.282795 = 0.3535 (rounded to 2 decimal places).
Therefore, the value of the test statistic is approximately 0.3535.