Question

Problem 1 - In a manufacturing process a random sample of 25 bolts has a mean bolt length of 10 inches with a
variance of .25 inches. Respond to each of the following questions being sure to show all of your work in the space(s) provided.
a. Should the researcher use t or z to compute this interval? Why so?
b. Compute the 95% confidence interval for the true mean length of the bolts in the overall population.
c. Compute the 99% confidence interval for the true mean length of the bolts in the overall population.
d. Weeks after you have completed the analysis required above, a person on the factory floor makes a comment suggesting
that the average bolt length is above 10.3 inches. You think he is mistaken. How sure are you that he has made an error?
Explain to him, in everyday English, why your opinion is so strong.

Answer 1

Problem 1

sample size n = 25

mean bolt length = = 10 inch

variance = ² = 0.25 inches

(a) Here sample variances are given so we will use the t test here

standard error of sample mean = sqrt(0.25/25) = 0.1 inch

(b) 95% confidence interval for the true mean length of the bolts in the overall population.

= +- Z_95% se₀

= 10 +- 1.645 * 0.1

= (9.8355 inch, 10.1645 inch)

(c)

99% confidence interval for the true mean length of the bolts in the overall population.

= +- Z_99% se₀

= 10 +- 2.576 * 0.1

= (9.7424 inch, 10.2576 inch)

(d) here the average bolt length is above 10.3 inches, so here whoever is suggesting is mistaken as there is very miniscule chance that bolt length would be above 10.3 inch. Here we are more than 99% sure that he has made an error.