Question

Assume that a simple random sample has been selected from a normally distributed population and test the given claim. Use the traditional method. Identify the null and alternative hypotheses, test statistic, critical value(s), and state the final conclusion that addresses the original claim.

A manufacturer makes ball bearings that are supposed to have a mean weight of 30 g. A retailer suspects that the mean weight is actually less than 30 g. The mean weight for a
random sample of 16 ball bearings is 29.2 g with a standard deviation of 4.2 g. At the 0.05 significance level, test the claim that the sample comes from a population with a mean

weight less than 30 g. Use the traditional method of testing hypotheses.

Answer 1

Solution:

The null and alternative hypotheses are:

$H_{0}:\mu=30$

$H_{a}:\mu<30$

Under the null hypothesis, the test statistic is:

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

$\large =\frac{29.2-30}{\frac{4.2}{\sqrt{16}}}$

$\large =\frac{-0.8}{1.05}$

$\large =-0.76$

Therefore, the test statistic is $\boldsymbol{t=-0.76}$ .

Now we have to find the left tailed critical value at 0.05 significance level for $df=n-1=16-1=15$

Using the t distribution table, we have:

$\boldsymbol{t_{critical}=-1.753}$

Conclusion: Since the test statistic does not lie outside the critical region, we, therefore, fail to reject the null hypothesis and conclude that there is not sufficient evidence to support the claim that the mean weight is less than 30 g.