Question

short answer. write the word or phrase that best completes each statement or answers the question.
assume that a simple random sample has been selected from a normally distributed population. find the test statistic,
p - value, critical value(s), and state the final conclusion.

1. test the claim that for the adult population of one town, the mean annual salary is given by \\( \mu = \\$ 30,000 \\). sample data are summarized as \\( n = 17, \overline { x } = \\$ 22,298 \\), and \\( s = \\$ 14,200 \\). use a significance level of \\( \alpha = 0.05 \\).

Explanation:

Step1: State the hypotheses

Let \(H_0:\mu = 30000\) (null hypothesis) and \(H_1:\mu
eq30000\) (alternative hypothesis).

Step2: Calculate the test statistic

The test statistic for a one - sample \(t\) - test is \(t=\frac{\bar{x}-\mu}{s/\sqrt{n}}\).
Substitute \(\bar{x} = 22298\), \(\mu = 30000\), \(s = 14200\), and \(n = 17\) into the formula:

$$t=\frac{22298 - 30000}{14200/\sqrt{17}}$$

$$t=\frac{- 7702}{14200/4.123}$$

$$t=\frac{-7702}{3444.1}$$

\(t\approx - 2.24\)

Step3: Find the degrees of freedom

Degrees of freedom \(df=n - 1=17-1 = 16\)

Step4: Find the critical values

For a two - tailed test with \(\alpha = 0.05\) and \(df = 16\), the critical values are \(t_{\alpha/2,df}=t_{0.025,16}\)
From the \(t\) - distribution table, \(t_{0.025,16}=\pm 2.120\)

Step5: Find the P - value

Using a \(t\) - distribution calculator or software with \(t=-2.24\) and \(df = 16\), the two - tailed \(P\) - value is \(P\approx0.04\)

Step6: State the conclusion

Since \(|t|=2.24>2.120\) (critical value) and \(P = 0.04<0.05\) (significance level), we reject the null hypothesis \(H_0\)

Answer:

Test statistic \(t\approx - 2.24\), critical values \(t=\pm2.120\), \(P\) - value \(\approx0.04\). Conclusion: Reject the null hypothesis \(H_0\)