Question

A melting point test of n D 10 samples of a binder used in manufacturing a rocket propellant resulted in x D 154:2 F. Assume that the melting point is normally distributed with D 1:5 F: Suppose you want to test the claim that the temperature is different from 155 F: Test the hypothesis at D 0:05:

Write down the hypothesis to be tested and show all the steps and calculations.

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\mu=155$

$H_1:\mu\not=155$

​This corresponds to a two-tailed test, for which a z-test for one mean, with known population standard deviation will be used.

Based on the information provided, the significance level is $\alpha=0.05,$ and the critical value for a two-tailed test is $z_c=1.96.$

The rejection region for this two-tailed test is$R=\{z:|z|>1.96\}.$

The z-statistic is computed as follows:

$$z=\dfrac{\bar{x}-\mu}{\sigma/\sqrt{n}}=\dfrac{154.2-155}{1.5/\sqrt{10}}$$

$$\approx-1.686548$$

Since it is observed that $|z|=1.686548<1.96=z_c,$ it is then concluded that the null hypothesis is not rejected.

Using the P-value approach:

The p-value is $p=2P(Z<-1.686548)=0.09169,$ and since $p=0.09169>0.05=\alpha,$ it is concluded that the null hypothesis is not rejected.

Therefore, there is not enough evidence to claim that the population mean $\mu$

is different than 155, at the $\alpha=0.05$ significance level.