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. Weekly Earnings of Women According to the Women's Bureau of the U.S. Department of Labor, the occupation with the highest median weekly earnings among women is pharmacist with median weekly earnings of dollar 1603. Based on the weekly earnings listed from a random sample of female pharmacists, can it be concluded that the median is less than dollar 1603 ? Use $\alpha=0.05 .$ $$ \begin{array}{lll}{1550} & {1355} & {1777} \\ {1430} & {1570} & {1701} \\ {2465} & {1655} & {1484} \\ {1429} & {1829} & {1812} \\ {1217} & {1501} & {1449}\end{array} $$

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.
Weekly Earnings of Women According to the Women's Bureau of the U.S. Department of Labor, the occupation with the highest median weekly earnings among women is pharmacist with median weekly earnings of dollar 1603. Based on the weekly earnings listed from a random sample of female pharmacists, can it be concluded that the median is less than dollar 1603 ? Use $\alpha=0.05 .$
$$
\begin{array}{lll}{1550} & {1355} & {1777} \\ {1430} & {1570} & {1701} \\ {2465} & {1655} & {1484} \\ {1429} & {1829} & {1812} \\ {1217} & {1501} & {1449}\end{array}
$$

Key Concepts

Decision Rule

The decision rule in hypothesis testing is the criterion used to decide whether to reject or fail to reject the null hypothesis. It is formulated based on the significance level and the critical value(s); if the test statistic falls within the rejection region, the decision rule leads to rejecting H?. Otherwise, there is insufficient evidence to reject the null hypothesis.

Test Statistic

The test statistic is a standardized value computed from sample data that quantifies the degree of departure from the null hypothesis. Its calculation depends on the data and the specific statistical test employed. By comparing this value to the critical value, one can assess the evidence against the null hypothesis.

Summary and Interpretation of Results

After computing the test statistic and comparing it against the critical value(s), the results are summarized by making a conclusion regarding the hypotheses. This summary explains whether the data provide enough evidence to support the alternative hypothesis or if the null hypothesis stands, making it an important final step in communicating the outcome of the hypothesis test.

Critical Value

A critical value is a point (or points) on the scale of the test statistic beyond which we reject the null hypothesis. It defines the boundary of the rejection region for the test and is determined by the significance level and the nature of the test (one-tailed or two-tailed). If the computed test statistic falls into this rejection region, the null hypothesis is rejected.

Claim Identification

Identifying the claim within the hypothesis testing framework involves determining which hypothesis represents the assertion made by the researcher or the claim stated in the problem context. This helps in interpreting the test results in terms of whether they support the claim or not.

Formulating Null and Alternative Hypotheses

The null hypothesis (H?) is a statement of no effect or no difference and is assumed true until evidence suggests otherwise, while the alternative hypothesis (H?) indicates the presence of an effect or a difference. Clearly stating these hypotheses is critical because they define the focus and direction of the statistical test, including whether the test is one-tailed or two-tailed.

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences or decisions about a population parameter based on sample data. This systematic approach involves formulating a null hypothesis, representing a default position, and an alternative hypothesis, which represents a claim or an effect to be tested. The process then uses a test statistic calculated from the sample data to determine whether there is sufficient evidence to reject the null hypothesis in favor of the alternative.

Significance Level (?)

The significance level, often denoted by ?, is a predetermined threshold for deciding whether to reject the null hypothesis. It represents the probability of committing a Type I error, i.e., rejecting a true null hypothesis. Setting ?, such as 0.05, governs how strong the evidence must be before concluding that the observed data is unlikely under H?.

Transcript

00:01 Okay, in question eight, it is believed that the medium income for women for a week is $1 ,603.

00:09 Using some data points, they want to test to see if it's, they believe it might be less than that.

00:15 So we start off with our step one, which you can see here, h .o.

00:20 Is equal to, median is equal to 1603.

00:26 And our alternate hypothesis, the median is going to be less than 1603.

00:32 And since we're using less than, that makes this a one -tailed test.

00:35 Okay, so just remember that as we go forward.

00:41 You'll see it come up when we find the critical value, which is in step two.

00:44 So now we move on to step two.

00:46 We're going to look at the critical value.

00:48 So our cv using table j with an n of 15, a significance level of 0 .05, and the fact that it's a one -tailed test, we get our critical value to become three.

01:00 All right, so our critical value is three.

01:02 Now we move on to step three, which is to determine what our test value is.

01:07 So again, remember, your test value is you're going to take each of the data points and compare it to the believed median.

01:14 If the data point is less than the median, it takes on a negative value.

01:18 If it's greater than the median, it takes on a positive value.

01:20 So if you take your data points, you can just kind of go across the way the data is listed in the problem...

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