The null and alternate hypotheses are: H0: π1=π2 H1: π1≠π2 A sample of 200 observation

step 1 So we're assuming that the two population proportions are equal and alternately that the populat

The null and alternate hypotheses are: H0: π1=π2 ...

The null and alternate hypotheses are: $$\begin{array}{l}H_{0}: \pi_{1}=\pi_{2} \\H_{1}: \pi_{1} \neq \pi_{2}\end{array}$$ A sample of 200 observations from the first population indicated that $X_{1}$ is $170 .$ A sample of 150 observations from the second population revealed $X_{2}$ to be $110 .$ Use the .05 significance level to test the hypothesis. a. State the decision rule. b. Compute the pooled proportion. c. Compute the value of the test statistic. d. What is your decision regarding the null hypothesis?

The null and alternate hypotheses are:
$$\begin{array}{l}H_{0}: \pi_{1}=\pi_{2} \\H_{1}: \pi_{1} \neq \pi_{2}\end{array}$$
A sample of 200 observations from the first population indicated that $X_{1}$ is $170 .$ A sample of 150 observations from the second population revealed $X_{2}$ to be $110 .$ Use the .05 significance level to test the hypothesis.
a. State the decision rule.
b. Compute the pooled proportion.
c. Compute the value of the test statistic.
d. What is your decision regarding the null hypothesis?

Key Concepts

Pooled Proportion

The pooled proportion is an aggregated estimate of the common proportion when comparing two independent samples under the assumption that their true proportions are equal under the null hypothesis. It is obtained by summing the successes from both samples and dividing by the total number of observations across the samples.

Hypothesis Testing

Hypothesis testing is a statistical procedure used to make inferences about populations by comparing sample data against a stated assumption or hypothesis. It involves formulating a null hypothesis, which represents a baseline or status quo claim, and an alternate hypothesis, which represents what one might conclude if there is sufficient evidence against the null hypothesis.

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis (H0) is a statement of no effect or no difference that is presumed to be true until evidence suggests otherwise. The alternative hypothesis (H1) contradicts the null, asserting that there is an effect or a difference. Deciding whether to reject the null hypothesis depends on the test statistic and the level of significance.

Significance Level

The significance level, often denoted by ?, is the threshold for rejecting the null hypothesis in a statistical test. It represents the probability of rejecting the null hypothesis when it is actually true, thus controlling the Type I error rate. A common significance level used in research is 0.05.

Test Statistic (Z-test)

The test statistic in a hypothesis test for two proportions, commonly a Z-test, quantifies how many standard errors the observed difference in sample proportions is from the null hypothesis of no difference. It uses the pooled proportion in its calculation to account for the assumption of equal population proportions under the null hypothesis.

Decision Rule

The decision rule is a predetermined criterion for deciding whether to reject the null hypothesis. It is based on the chosen significance level and the distribution of the test statistic (e.g., the critical values from the standard normal distribution for a Z-test). If the calculated test statistic exceeds the critical value, the null hypothesis is rejected.

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