Question

A random sample of n= 21 items is taken, resulting in a sample mean x¯= 17.0 with a sample standard deviation of s= 4.9. Assume x is normally distributed and use this information and α 0.05 test the following hypotheses.

H0: μ = 16

Ha: μ ≠ 16 

Determine the critical value tc and t score value t based on the sample data.

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\mu=16$

$H_1:\mu\not=16$

This corresponds to a two-tailed test, for which a t-test for one mean, with unknown population standard deviation, using the sample standard deviation, will be used.

Based on the information provided, the significance level is $\alpha = 0.05,$ $df=n-1=20$ degrees of freedom, and the critical value for a two-tailed test is $t_c=2.085963.$

The rejection region for this two-tailed test is $R = \{t: |t| > 2.085963\}.$

The t-statistic is computed as follows:

$$t=\dfrac{\bar{X}-\mu}{s/\sqrt{n}}=\dfrac{17-16}{4.9/\sqrt{21}}=0.935220$$