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. Ages at First Marriage for Women The median age at first marriage in 2014 for women was 27 years- the highest it has ever been. A random sample of women's ages (in years) from recently applied for marriage licenses resulted in the following set of ages. At $\alpha=0.05,$ is there sufficient evidence that the median is not 27 years? $$ \begin{array}{lllll}{34.6} & {31.2} & {28.9} & {28.4} & {24.3} \\ {29.8} & {25.9} & {21.4} & {25.1} & {26.2} \\ {28.3} & {30.6} & {35.6} & {34.2} & {34.1}\end{array} $$

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.
Ages at First Marriage for Women The median age at first marriage in 2014 for women was 27 years- the highest it has ever been. A random sample of women's ages (in years) from recently applied for marriage licenses resulted in the following set of ages. At $\alpha=0.05,$ is there sufficient evidence that the median is not 27 years?
$$
\begin{array}{lllll}{34.6} & {31.2} & {28.9} & {28.4} & {24.3} \\ {29.8} & {25.9} & {21.4} & {25.1} & {26.2} \\ {28.3} & {30.6} & {35.6} & {34.2} & {34.1}\end{array}
$$

Key Concepts

Critical Value

A critical value is the threshold value that the test statistic must exceed in order for the null hypothesis to be rejected. It is determined based on the chosen significance level (alpha) and the sampling distribution of the test statistic under the null hypothesis.

Decision Rule and Conclusion

This involves comparing the computed test statistic to the critical value. If the test statistic falls in the rejection region defined by the critical value, the null hypothesis is rejected. Finally, the statistical conclusion is summarized, relating the result of the test back to the context of the research question.

Test Statistic

The test statistic is a numerical value calculated from sample data used to decide whether to reject the null hypothesis. Its computation involves comparing observed data to what would be expected under the null hypothesis, such as counting the number of observations above or below the hypothesized median in a sign test.

Nonparametric Test Methods

Often when testing hypotheses about medians, nonparametric methods such as the sign test are used because they do not assume a specific data distribution. These tests are particularly useful when the data does not meet the assumptions required for parametric tests, such as normality.

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis (H0) typically states that there is no difference or effect—in this context, that the median is a specific value. The alternative hypothesis (Ha) contests the null and posits that the median is different from the specified value. The testing process begins by clearly stating these hypotheses and identifying which one is being claimed as true.

Hypothesis Testing

This is a systematic method used in statistics to decide whether there is enough evidence in a sample of data to infer that a certain condition holds for the entire population. It involves formulating a null hypothesis, which represents the status quo or a statement of no effect, and an alternative hypothesis, representing the claim that the study is testing for.

Transcript

00:01 We're going to perform a sign test for question number five, which talks about the median age of marriage in women.

00:08 So the claim is that the median age is 27 back in, i believe the question talked about 2014.

00:18 But so we'll start.

00:19 So we're going to go step one.

00:26 So step one, according to right here, is we want to state the hypothesis.

00:32 So when stating the hypothesis, we are going to start with our null hypothesis, which is going to be that the median is equal to 27.

00:49 Now, the ultimate hypothesis, which is stating that it's not true that the median is 27.

01:00 Okay, so there's your hypothesis that we're going to be working with.

01:04 Right.

01:06 Your step two is to find the critical value.

01:15 So the critical value when n is going to be less than or equal to 25 is we're going to look it up on table g.

01:23 And we get a critical value that's going to be equal to.

01:29 Right.

01:30 And so now what we want to do is we want to move on to step four.

01:40 Actually step three step four step three and we want to find what the test value is so in determining the test value we're going to have to take a look at all of the data points and we're going to have to create signs for each one and we're going to basically if the data value is bigger than the median we're going to give it a plus sign and if it's small than the median we're going to give it a minus sign so when we go in the order in which the numbers were given to us in the problem, again, we're comparing everything to 27.

02:24 So 36 .34 .6 is greater than.

02:28 So it goes down as a plus.

02:30 We'll put the pluses down as a green.

02:32 31 .2 is a plus...

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