Math / Data & Statistics

QuestionUse a t-test to test the claim about the population mean $\mu$ at the given level of significance $\alpha$ using the given sample statistics. Assume the population is normally distributed. Claim: $\mu \geq 8300 ; \alpha=0.10$ Sample statistics: $\bar{x}=8100, s=460, n=25$
What are the null and alternative hypotheses? A. $\mathrm{H}_{0}: \mu \neq 8300$ B. $H_{0}: \mu=8300$ $H_{a}: \mu=8300$ $H_{a}: \mu \neq 8300$ C. $H_{0}: \mu \geq 8300$ D. $H_{0}: \mu \leq 8300$ $H_{a}: \mu<8300$ $H_{a}: \mu>8300$
What is the value of the standardized test statistic? The standardized test statistic is $\square$ (Round to two decimal places as needed.)

Studdy Solution

STEP 1

1. The population is normally distributed.
2. We are using a t-test to evaluate the claim about the population mean $\mu$ .
3. The sample size $n = 25$ is less than 30, so a t-test is appropriate.
4. The sample mean $\bar{x} = 8100$ , the sample standard deviation $s = 460$ , and the level of significance $\alpha = 0.10$ .

STEP 2

1. Determine the null and alternative hypotheses.
2. Calculate the standardized test statistic.

STEP 3

Determine the null and alternative hypotheses based on the claim $\mu \geq 8300$ .
- The null hypothesis $H_0$ represents the statement of no effect or status quo. For this claim, it would be $H_0: \mu \geq 8300$ . - The alternative hypothesis $H_a$ represents the statement we are testing against the null hypothesis. Since we are testing if $\mu$ is less than 8300, it would be $H_a: \mu < 8300$ .
Thus, the correct option is: D. $H_0: \mu \geq 8300$ , $H_a: \mu < 8300$

STEP 4

Calculate the standardized test statistic using the formula for the t-statistic:
$t = \frac{\bar{x} - \mu}{s/\sqrt{n}}$
Substitute the given values:
$t = \frac{8100 - 8300}{460/\sqrt{25}}$
Calculate the denominator:
$\frac{460}{\sqrt{25}} = \frac{460}{5} = 92$
Calculate the t-statistic:
$t = \frac{8100 - 8300}{92} = \frac{-200}{92} \approx -2.17$
The standardized test statistic is:
$\boxed{-2.17}$

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