Question

A trucking company continuously monitors the tread remaining on its fleet of truck tires. The firm knows that standard deviation is 0.1 cm. Suppose a sample of 80 tires were obtained, and the mean tread was 2.2 cm. Construct a 99% confidence interval. The company's operations manager knows that flat tires occur more often when the tread is below 1.0 cm. At what sample mean (and higher) should the company reasonably expect minimal flat tires, based on a 95% confidence interval?

Answer 1

Standard error of mean = = 0.1 / = 0.01118034

Z value for 99% confidence interval is  2.576

99% confidence interval for mean tread is,

(2.2 - 2.576 * 0.01118034 ,  2.2 + 2.576 * 0.01118034)

(2.17 cm,  2.23 cm)

Let x be the sample mean.

For 95% confidence interval, the company reasonably expect minimal flat tires if the 1.0 cm lies within the 95% confidence interval.

Z value for 95% confidence interval is 1.645

The lower and upper limits of 95% confidence interval are,

(x - 1.645 * 0.01118034, x + 1.645 * 0.01118034)

(x - 0.0184, x + 0.0184)

For minimal flat tires, the lower limit of the interval should be greater than 1.0 cm

x - 0.0184 > 1

x > 1 + 0.0184

x > 1.0184 cm

The sample mean should be at least 1.0184 cm to reasonably expect minimal flat tires based on a 95% confidence interval.