Calculating Power Consider a hypothesis test of the claim...

Calculating Power Consider a hypothesis test of the claim that the Ericsson method of gender selection is effective in increasing the likelihood of having a baby girl, so that the claim is $p>0.5$. Assume that a significance level of $\alpha=0.05$ is used, and the sample is a simple random sample of size $n=64$. a. Assuming that the true population proportion is $0.65$, find the power of the test, which is the probability of rejecting the null hypothesis when it is false. (Hint: With a $0.05$ significance level, the critical value is $z=1.645$, so any test statistic in the right tail of the accompanying top graph is in the rejection region where the claim is supported. Find the sample proportion $\hat{p}$ in the top graph, and use it to find the power shown in the bottom graph.) b. Explain why the green-shaded region of the bottom graph represents the power of the test.

Calculating Power Consider a hypothesis test of the claim that the Ericsson method of gender selection is effective in increasing the likelihood of having a baby girl, so that the claim is $p>0.5$. Assume that a significance level of $\alpha=0.05$ is used, and the sample is a simple random sample of size $n=64$.
a. Assuming that the true population proportion is $0.65$, find the power of the test, which is the probability of rejecting the null hypothesis when it is false. (Hint: With a $0.05$ significance level, the critical value is $z=1.645$, so any test statistic in the right tail of the accompanying top graph is in the rejection region where the claim is supported. Find the sample proportion $\hat{p}$ in the top graph, and use it to find the power shown in the bottom graph.)
b. Explain why the green-shaded region of the bottom graph represents the power of the test.

Key Concepts

Type II Error

A Type II error occurs when the test fails to reject a false null hypothesis. The probability of committing a Type II error is inversely related to the power of the test; hence, higher power means a lower chance of a Type II error.

Graphical Representation in Hypothesis Testing

Graphical illustrations in hypothesis testing, such as shaded regions under the sampling distribution curve, visually represent areas corresponding to critical values and probabilities. The specific shading indicating the power of a test reflects the probability of the test statistic falling into the rejection region under the alternative hypothesis.

Sampling Distribution of the Test Statistic

The sampling distribution of the test statistic, such as the sample proportion, describes how the statistic varies from sample to sample. Understanding this distribution, including its mean and standard error, is essential for determining probabilities associated with observed test statistics under both the null and alternative hypotheses.

Significance Level and Critical Value

The significance level, denoted as alpha, represents the threshold probability for rejecting the null hypothesis when it is actually true. In the context of hypothesis testing, the critical value corresponds to this significance level and defines the boundary of the rejection region in the test's distribution.

Hypothesis Testing Framework

Hypothesis testing involves formulating a null hypothesis and an alternative hypothesis, then using sample data to assess the strength of evidence against the null hypothesis. Decisions are made based on whether the test statistic falls within the rejection region defined by the chosen significance level.

Statistical Power

Statistical power is the probability that a statistical test will correctly reject a false null hypothesis. It measures the test's ability to detect an effect or difference when one actually exists and is influenced by factors such as sample size, effect size, and significance level.

Transcript

00:01 In this problem, we need to find the power of this test, which is the probability of rejecting the null hypothesis when it is false.

00:14 Which recall is a correct conclusion.

00:19 We want to reject the null hypothesis when it is false.

00:24 First, i like to write down the alternative hypothesis, which we call h1.

00:28 The alternative hypothesis is the hypothesis that we're trying to test or that the researcher wants to quote unquote prove.

00:39 That would be that p, the true proportion, is greater than 0 .5.

00:47 So the null hypothesis when we're using proportions is always going to be the equal sign.

00:56 So the true proportion p is equal to 0 .5.

01:01 So there's our null hypothesis and there's our alternative hypothesis.

01:10 So then remember that the power is the probability of rejecting the null hypothesis when it is false.

01:22 So this would be a correct conclusion because if the null hypothesis is false, we want to reject it.

01:28 The first thing we're going to do, we're told we have a little hint here.

01:33 We have an alpha equals 0 .05.

01:38 Remember alpha is the area in green here.

01:42 Alpha corresponds to an area, so the area in green is 0 .05.

01:47 And we are told that this value here is z equals 1 .645.

01:54 And we are asked to find the sample proportion p hat in the top graph that corresponds to this value of 0 .645.

02:03 So we want p hat, which is the sample proportion that corresponds to that value of z.

02:13 And then we're going to use that value once we calculate it.

02:16 We'll use that value here in the bottom graph.

02:20 And finally, that will help us to solve for the power.

02:24 What we will do next is find value of p hat.

02:30 How are we going to get the value of p hat? well we can actually solve for it.

02:34 On page 389 there's a table at the top and it tells us that for proportion p, the sampling distribution is a normal distribution.

02:46 And that the test statistic is z equals p hat minus p divided by p times q divided by n all under the square root.

02:59 From page 389 in my textbook.

03:03 So we have z.

03:04 We know what p is, what value do we use here for p.

03:08 We actually use this value from the null hypothesis.

03:11 So we will insert 0 .5 for p.

03:15 And then what is q? q is always 1 minus p.

03:19 So in this case q is going to be 1 minus 0 .5, which is 0 .5.

03:26 And then in, we were given the problem to be 64 so we have everything that we need we just need to solve for p hat solve for p hat all right so to solve and isolate p hat first thing we should do is multiply both sides by the denominator on the left hand side we'll get z times the square root of p q over n equal sign and then on the right hand side we will get just the numerator p hat minus p okay that was by multiplying the denominator on both sides now we just want to isolate p hat so we have just one more step and that is to add p to both sides so when we do that we get p plus z square root pq over n on the left hand side equals p hat.

04:36 Okay, so now we've isolated p hat.

04:39 This will be our formula for finding p hat.

04:42 So we're one step closer.

04:44 Let's make some room here.

04:46 All right, p hat is going to be, i'm just going to rewrite the formula, p plus z times pq over in.

04:59 So let's just plug in everything that we know.

05:01 So p, remember, we're going to use the value from the null hypothesis.

05:08 So that's going to be 0 .5.

05:12 Plus, what is z? z we are given, and that is over here, 1 .645.

05:22 Let's plug that in, 1 .645 times the square root of p times q over n.

05:31 P is 0 .5.

05:34 We got that from the null hypothesis.