For Exercises 5 through 20, perform these steps. a. State...

For Exercises 5 through $20,$ perform these steps. a. State the hypotheses and identify the claim. b. Find the critical value(s). c. Compute the test value. d. Make the decision. e. Summarize the results. Use the traditional method of hypothesis testing unless otherwise specified. Annual Incomes for Men The U.S. median annual income for men in 2014 (in constant dollars) was dollar 35,642. A random sample of recent male college graduates indicated the following incomes. At the 0.05 level of significance, test the claim that the median is more than dollar 35,642.

For Exercises 5 through $20,$ perform these steps.
a. State the hypotheses and identify the claim.
b. Find the critical value(s).
c. Compute the test value.
d. Make the decision.
e. Summarize the results.
Use the traditional method of hypothesis testing unless otherwise specified.
Annual Incomes for Men The U.S. median annual income for men in 2014 (in constant dollars) was dollar 35,642. A random sample of recent male college graduates indicated the following incomes. At the 0.05 level of significance, test the claim that the median is more than dollar 35,642.

Key Concepts

Summary of Results

After applying the test procedures, the summary of results interprets the statistical findings in context. This includes stating whether there is sufficient evidence to support the claim based on the decision from the hypothesis test, and explaining what that means in real-world terms.

Test Statistic

The test statistic is a standardized value calculated from sample data that measures the degree of departure of the sample statistic from the null hypothesis parameter. Its computed value is compared against the critical value to decide whether to reject H?.

Decision Rule

The decision rule is a guideline established before analyzing the data. It involves comparing the test statistic to the critical value(s); if the test statistic falls within the rejection region, the null hypothesis is rejected, otherwise, it is not. This ensures objectivity in the hypothesis evaluation.

Significance Level

The significance level, often denoted by ?, is the probability threshold set by the researcher for rejecting the null hypothesis. It quantifies the risk of committing a Type I error (rejecting a true null hypothesis). Common levels used are 0.05, 0.01, or 0.10.

Null and Alternative Hypothesis

In hypothesis testing, the null hypothesis (H?) represents the default or status quo assumption about a population parameter, while the alternative hypothesis (H?) represents a claim or effect that the researcher wishes to test. The alternative hypothesis is generally what a claim or theory asserts, such as stating that a population parameter is greater than a certain value.

Hypothesis Testing Procedures

This concept refers to the systematic approach used to decide whether there is enough evidence in a sample to support a specific claim about a population parameter. The process typically involves stating the null and alternative hypotheses, selecting a significance level, determining the appropriate test statistic and its critical value(s), making a decision based on comparison of the test statistic to the critical value, and finally summarizing the results in context.

Critical Value

The critical value is a point (or points) on the scale of the test statistic beyond which the null hypothesis is rejected. It is derived from the probability distribution associated with the test statistic based on the chosen significance level and the nature (one-tailed or two-tailed) of the test.

Transcript

00:01 Okay, so for question number seven, question number seven talks about the median income of men in 2014, and is believed that the median income is 35 ,642.

00:13 And they're testing based on some of the data, 13 data points, 13 salaries.

00:18 They want to test to see if the median income is actually greater than that number.

00:25 All right, so we have our hypothesis for step one.

00:29 We have our first hypothesis, the median income, which equals 35 ,642.

00:35 That's here.

00:36 And then the belief in the question that we're looking to kind of show is that the median income is actually really greater than 35 ,000 is 20.

00:45 So we move on to step two.

00:47 Step two, we have to come up with our critical value.

00:50 So our critical value using table j, when n is equal to 13, all right, n is equal to 13, alpha level being 0 .05.

00:59 And it's a one -tail test.

01:00 Now, it's a one -tail test this time because if you take a look at our h -1, our ultimate hypothesis, this one here, we used the greater -than -simble.

01:12 So we're looking at only one side.

01:14 We're not looking at not equal to, which would be two sides.

01:16 So it's a one -tail test.

01:20 And when you take all this into account, any is equal to 13, your alpha level 0 .05, and it's a one -tail test, the critical value is three for the question.

01:28 Okay.

01:30 Next step, we're going to go find our test value.

01:32 So as we have done before, the test value is found by taking a look at each of the income spots and in the data.

01:42 So the first data point, for example, is 35 ,000...

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