Math / Data & Statistics

QuestionFor the following claim, find the null and alternative hypotheses, test statistic, critical value, and draw a conclusion. Assume that a simple random sample has been selected from a normally distributed population. Answer parts a-d.
Claim: The mean IQ score of statistics professors is greater than 128.
Sample data: $n=17, \bar{x}=131, s=11$ . The significance level is $\alpha=0.05$ . (7) Click the icon to view a table of critical t-values. $\mathrm{t}=1.124$ (Round to three decimal places as needed.) c. Find the critical value using a t-distribution table.
The critical value is $\square$ (Round to three decimal places as needed.)

Studdy Solution

STEP 1

1. The sample is a simple random sample from a normally distributed population.
2. The sample size $n = 17$ .
3. The sample mean $\bar{x} = 131$ .
4. The sample standard deviation $s = 11$ .
5. The significance level $\alpha = 0.05$ .

STEP 2

1. Define the null and alternative hypotheses.
2. Calculate the test statistic.
3. Determine the critical value using a t-distribution table.
4. Draw a conclusion based on the test statistic and critical value.

STEP 3

Define the null and alternative hypotheses.
- Null Hypothesis ( $H_0$ ): The mean IQ score of statistics professors is equal to 128. $H_0: \mu = 128$
- Alternative Hypothesis ( $H_a$ ): The mean IQ score of statistics professors is greater than 128. $H_a: \mu > 128$

STEP 4

Calculate the test statistic using the formula for the t-test:
$t = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}}$
where $\mu_0 = 128$ , $\bar{x} = 131$ , $s = 11$ , and $n = 17$ .
$t = \frac{131 - 128}{\frac{11}{\sqrt{17}}}$
Calculate the value:
$t = \frac{3}{\frac{11}{\sqrt{17}}} \approx 1.124$

STEP 5

Determine the critical value using a t-distribution table.
- Degrees of freedom ( $df$ ) is $n - 1 = 17 - 1 = 16$ . - Significance level $\alpha = 0.05$ for a one-tailed test.
Using a t-distribution table, find the critical value for $df = 16$ and $\alpha = 0.05$ .
The critical value is approximately $t_{critical} = 1.746$ .

STEP 6

Draw a conclusion based on the test statistic and critical value.
- Compare the test statistic $t = 1.124$ with the critical value $t_{critical} = 1.746$ .
Since $1.124 < 1.746$ , we fail to reject the null hypothesis.
Conclusion: There is not enough evidence to support the claim that the mean IQ score of statistics professors is greater than 128 at the $\alpha = 0.05$ significance level.

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