Final answer:
To test the null hypothesis that the population mean weight of the bolts is 82.9 grams against the alternative hypothesis that it differs from 82.9 grams, we use a t statistic. The value of the appropriate test statistic is 103.33 and the p-value of the test is less than 0.001. Therefore, we reject the null hypothesis and conclude that the population mean weight differs from 82.9 grams.
Step-by-step explanation:
(a) To test the null hypothesis, we can use a t statistic because the population standard deviation is not known and the sample size is relatively small (n=100). Using a t statistic allows us to account for the uncertainty associated with estimating the population standard deviation from the sample.
(b) The value of the appropriate test statistic can be calculated using the formula:
t = (sample mean - hypothesized mean) / (sample standard deviation / sqrt(sample size))
t = (84.14 - 82.9) / (0.12 / sqrt(100))
t = 1.24 / 0.012 = 103.33
(c) The p-value of the test can be calculated by finding the probability of observing a test statistic as extreme as 103.33 (in either direction) under the null hypothesis. Since the alternative hypothesis is two-sided, we need to find the area in both tails of the t-distribution.
Using the t-distribution table or a statistical software, we find that the p-value is less than 0.001.
(d) At the 0.1% level of significance, since the p-value is less than 0.001, we reject the null hypothesis. There is strong evidence to suggest that the population mean weight of this type of bolt differs from 82.9 grams.