The null and alternate hypotheses are: H0: π1 ≤π2 H1: π1>π2 A sample of 100 observations from th

The null and alternate hypotheses are: H0: π1 ≤π2 H1: π1>π2...

The null and alternate hypotheses are: $H_{0}: \pi_{1} \leq \pi_{2}$ $H_{1}: \pi_{1}>\pi_{2}$ A sample of 100 observations from the first population indicated that $X_{1}$ is $70 .$ A sample of 150 observations from the second population revealed $X_{2}$ to be $90 .$ 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:
$H_{0}: \pi_{1} \leq \pi_{2}$
$H_{1}: \pi_{1}>\pi_{2}$
A sample of 100 observations from the first population indicated that $X_{1}$ is $70 .$ A sample of 150 observations from the second population revealed $X_{2}$ to be $90 .$ 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

Null and Alternative Hypotheses

The null hypothesis (H0) and the alternative hypothesis (H1) are statements that guide the testing process. The null hypothesis typically represents the status quo or a statement of no effect or no difference, while the alternative hypothesis represents what the researcher expects or is testing for. In the context of comparing two proportions, the hypotheses set the framework for determining whether one population proportion is statistically significantly greater than another.

Significance Level

The significance level, often denoted by ?, is the pre-determined probability threshold used for rejecting the null hypothesis. It represents the risk of making a Type I error—rejecting the null hypothesis when it is actually true. A common significance level is 0.05, which means there is a 5% risk of erroneously rejecting the null hypothesis.

Pooled Proportion

The pooled proportion is a combination of the success counts from two independent samples when the null hypothesis assumes that the two population proportions are equal. It serves as a common estimate of the shared proportion under H0, and is used to compute the standard error of the difference in proportions for the test statistic.

Test Statistic

The test statistic in the context of comparing two proportions often takes the form of a z-score, which standardizes the difference between the observed sample proportions against the null hypothesis value, using the pooled proportion to estimate the standard error. This statistic is then compared with critical values from the standard normal distribution for the purpose of decision making.

Decision Rule

The decision rule is a pre-specified criterion that determines whether to reject or fail to reject the null hypothesis. It involves comparing the computed test statistic to critical values corresponding to the set significance level. In the case of a one-tailed test, if the test statistic exceeds the critical value (or falls into the rejection region), the null hypothesis is rejected in favor of the alternative hypothesis.

Transcript

00:01 So we're assuming that the proportion 1 is less than or equal to proportion 2.

00:07 And alternately, that proportion 1 is greater than proportion 2.

00:12 So we're doing a 1 -tel test.

00:14 And we have our first proportion being that 70 out of 100.

00:20 And our second proportion is the 90 out of 150.

00:26 And question a asks us to determine what our decision rule, and we're using a 5 % significance level with one tail.

00:36 So we're going to put all 5 % in that upper tail, and it's a z value, and we would reject the null.

00:45 If we are above, and let's look at 5%, it's going to be that 1 .645, pretty sure.

00:51 Yep, 1 .645.

00:53 So if we are higher than that value for our test statistic, we're going to reject the null.

00:58 Now part b wants us to find that pooled p, which they labeled p sub c.

01:03 And so our pooled p is to sum up these two, 70 plus 90, and then 100 plus 150.

01:12 And i can write, apparently not very well, 150.

01:18 There we go.

01:19 So we get 160 out of 250, and that proportion comes out to be as a decimal, as this one is naturally 0 .7, and this one is 0 .6, and this pooled p is 0 .64.

01:38 So part c, we need to figure out what our test statistic is.

01:42 So we're going to take the difference between our two proportions, 0 .7 minus 0 .6...

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