Question

A simple random sample of size n=15 is drawn from a population that is normally distributed. The sample mean is found to be x =18.8 and the sample standard deviation is found to be s=6.3. Determine if the population mean is different from 24 at the alpha equals 0.01 level of significance.
(a) Determine the null and alternative hypotheses.
b) Calculate the P-value.
(c) State the conclusion for the test.
(d) State the conclusion in context of the problem.

Answer 1

Answer:

(a) Determine the null and alternative hypotheses.

$H_{0}:\mu=24$

$H_{a}:\mu \neq 24$

b) Calculate the P-value.

We need to find the test statistic in order to find the P-value. The test statistic is:

$t=\frac{\bar{x}-\mu}{\frac{s}{\sqrt{n}} }$

     $=\frac{18.8-24}{\frac{6.3}{\sqrt{15}} }$

     $=-3.20$

Now using the t distribution, the P-value is:

$P-value=P(t<-3.20)+P(t>3.20)$

            $=0.0032+0.0032$

            $=0.0064$

(c) State the conclusion for the test.

Since the P-value is less than the significance level, we therefore, reject the null hypothesis.

(d) State the conclusion in context of the problem.

The null hypothesis is rejected at 0.01 significance level, we therefore, have sufficient evidence to support the claim that the population mean is different from 24 at 0.01 significance level.