The Employment and Training Administration reported that the...

The Employment and Training Administration reported that the U.S. mean unemployment insurance benefit was $$\$ 238$$ per week (The World Almanac, 2003). A researcher in the state of Virginia anticipated that sample data would show evidence thar the mean weekly unemployment insurance benefit in Virginia was below the national average. a. Develop appropriate hypotheses such that rejection of $H_0$ will support the researcher's contention. b. For a sample of 100 individuals, the sample mean weekly unemployment insurance benefit was $$\$ 231$$ with a sample standard deviation of $\$ 80$. What is the p-value? c. At $a=.05$, what is your conclusion? d. Repear the preceding hypothesis test using the critical value approach.

The Employment and Training Administration reported that the U.S. mean unemployment insurance benefit was $$\$ 238$$ per week (The World Almanac, 2003). A researcher in the state of Virginia anticipated that sample data would show evidence thar the mean weekly unemployment insurance benefit in Virginia was below the national average.
a. Develop appropriate hypotheses such that rejection of $H_0$ will support the researcher's contention.
b. For a sample of 100 individuals, the sample mean weekly unemployment insurance benefit was $$\$ 231$$ with a sample standard deviation of $\$ 80$. What is the p-value?
c. At $a=.05$, what is your conclusion?
d. Repear the preceding hypothesis test using the critical value approach.

Key Concepts

Significance Level

The significance level (denoted by ?) is the threshold probability below which the null hypothesis is rejected. It represents the maximum acceptable risk of committing a Type I error, which is rejecting a true null hypothesis. In this context, an ? of 0.05 is used to decide whether the evidence is strong enough to support the researcher's claim.

Hypothesis Formulation

This concept involves stating the null hypothesis (H?) and the alternative hypothesis (H?) based on the research question. The null hypothesis typically reflects no difference or change—in this case, that the Virginia unemployment benefits are equal to the national mean—while the alternative represents the researcher's claim that they are lower. It sets the framework for testing a statistical claim.

One-Sample t-Test

The one-sample t-test is used to determine if the sample mean differs significantly from a known or hypothesized population mean when the population standard deviation is unknown and the sample size is relatively small or moderate. It involves computing a test statistic that measures the difference between the sample mean and the hypothesized population mean relative to the sample’s variability.

p-value

The p-value is the probability of observing a test statistic as extreme as, or more extreme than, the one computed from the sample data, assuming the null hypothesis is true. It quantifies the evidence against the null hypothesis; a smaller p-value indicates stronger evidence to reject H?.

Critical Value Approach

The critical value approach compares the calculated test statistic to a threshold value (critical value) from the statistical distribution corresponding to the chosen significance level. If the test statistic falls into the rejection region defined by the critical value (e.g., in the lower tail for a one-tailed test), the null hypothesis is rejected, supporting the alternative hypothesis.

Transcript

00:01 Question 27, we're told that the weekly average unemployment benefit in the u .s.

00:05 Is $238.

00:10 And we're also told that a researcher in virginia state suspects that the state average there is less than $238.

00:17 So for part a, we're asked to develop a hypothesis test for this situation.

00:22 So we can describe the virginia state researchers view as the alternative hypothesis that mu is less than 238, and therefore the no hypothesis is mu is greater than or equal to 238.

00:40 So that's your hypothesis test.

00:44 For part b, we're told that we have a sample from virginia state of size 100, and the average of the sample is $231, and the sample standard deviation is $80.

00:59 And we're asked what the p value is.

01:02 So we're not given the population standard deviation, so we can say that sample averages are distributed, according to the t distribution.

01:11 So t is our test statistic and that is equal to the sample average minus the population average over the sample standard deviation divided by the square root of the sample size.

01:38 And that comes out to minus 0 .88.

01:46 And then we're asked to calculate the p value.

01:49 So just to show you what's happening here.

01:54 So this is the t distribution, it's symmetrical.

02:00 And this is a lower tailed test because the alternative hypothesis suggests that that the mean is lower than the no hypothesis.

02:08 So basically we have a test statistic that comes out to minus 0 .88.

02:16 And we're basically asking the question, does it fall in some critical region in the lower tail such that it's low enough or extreme enough to reject the no hypothesis? so the critical region would be defined by alpha, which we haven't been given yet.

02:36 And so the p value is the area to the left of our test statistics.

02:42 So the red area.

02:48 So it amounts to comparing the p value to the alpha value to see which is smaller.

02:53 If the p value is smaller, then our test statistic is more extreme and lower, and which would lead us to rejecting the no hypothesis.

03:03 But of course the t table gives upper tail values, so we're looking at areas in the upper tail...

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