Question

A manufacturer of automobile batteries claim that at least 80% of the batteries
that it produces will last 36 months. A consumers’ advocate group wants to evaluate this
longevity claim and selects a random sample of 28 batteries to test. The following data indicate
the length of time (in months) that each of these batteries lasted (i.e., performed properly
before failure): 42.3, 39.6, 25.0, 56.2, 37.2, 47.4, 57.5, 39.3, 39.2, 47.0, 47.4, 39.7, 57.3, 51.8,
31.6, 45.1, 40.8, 42.4, 38.9, 42.9, 34.1, 49.0, 41.5, 60.1, 34.6, 50.4, 30.7, 44.1. Now, we would
like to test, at a significance level of 0.05, if there is a significant evidence that less than 80%
of the batteries will last at least 36 months? Conduct and conclude the test.

Answer 1

Solution:

We have to use the one-sample z-test for the proportion since np > 5 and n(1-p) > 5

The null and alternative hypotheses are:

Under the null hypothesis, the test statistic is:

Where:

is the sample proportion of batteries that last at least 36 months.

  

  

Now the p-value is:

Now using the standard normal table, we have:

Conclusion: Since the p-value is greater than the significance level, we, therefore, fail to reject the null hypothesis and there is not sufficient evidence to conclude that less than 80% of the batteries will last at least 36 months.