Math / Data & Statistics

QuestionQuestion 1 1 pts
A random sample of size 20 drawn from a normal population yielded the following results: $\bar{x}=49.2, s=1.33$ .
In testing: $H_{0}: \mu=50$ versus $H_{1}: \mu \neq 50$ at a 0.01 level of significance, the decision is to: not reject the null hypothesis. reject the null hypothesis.

Studdy Solution

STEP 1

1. The sample size is $n = 20$ .
2. The sample mean is $\bar{x} = 49.2$ .
3. The sample standard deviation is $s = 1.33$ .
4. The null hypothesis is $H_0: \mu = 50$ .
5. The alternative hypothesis is $H_1: \mu \neq 50$ .
6. The level of significance is $\alpha = 0.01$ .
7. The population is normally distributed.

STEP 2

1. Calculate the test statistic.
2. Determine the critical value(s) for the test.
3. Make a decision to reject or not reject the null hypothesis based on the test statistic and critical value(s).

STEP 3

Calculate the test statistic using the formula for the t-test:
$t = \frac{\bar{x} - \mu}{s / \sqrt{n}}$
Substitute the given values:
$t = \frac{49.2 - 50}{1.33 / \sqrt{20}}$
Calculate the value:
$t \approx \frac{-0.8}{0.2975} \approx -2.69$

STEP 4

Determine the critical value(s) for a two-tailed test at $\alpha = 0.01$ with $n - 1 = 19$ degrees of freedom.
Using a t-distribution table or calculator, find the critical values:
$t_{\text{critical}} \approx \pm 2.861$

STEP 5

Compare the calculated test statistic to the critical values:
Since $-2.69$ is greater than $-2.861$ and less than $2.861$ , it falls within the range of not rejecting the null hypothesis.
Decision: Not reject the null hypothesis.
The decision is to:
$\text{Not reject the null hypothesis.}$

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms