Question

I want to know how long basketball shoes last for people on average. I rubbed a dusty coffee pot and a genie popped out to tell me that the standard deviation for the lifespan of all basketball shoes is 4.2 months. I polled 1225 people at the PEIF (before it closed) on how long their shoes lasted, and I got an average of 27 months.

(a) What is the 95% confidence interval for the population mean lifespan of a basketball shoe?

(b) What is the 99.7% confidence interval for the population mean lifespan of a basketball shoe?

(c) Draw appropriate shaded bell curves to show the difference between these.

Answer 1

Solution:

a) The 95% confidence interval for population mean is given as follows:

Where, x̄ is sample mean, σ is population standard deviation, n is sample size and Z_(0.05/2) is critical z-value to construct 95% confidence interval.

We have, x̄ = 27 months, σ = 4.2 months, n = 1225

Using Z-table we get, Z_(0.05/2) = 1.96

Hence, the 95% confidence interval for the population mean lifespan of a basketball shoe is,

The 95% confidence interval for the population mean lifespan of a basketball shoe is (26.7649 months, 27.2352 months).

b) The 99.7% confidence interval for population mean is given as follows:

Where, x̄ is sample mean, σ is population standard deviation, n is sample size and Z_(0.003/2) is critical z-value to construct 99.7% confidence interval.

We have, x̄ = 27 months, σ = 4.2 months, n = 1225

Using Z-table we get, Z_(0.003/2) = 2.97

Hence, the 99.7% confidence interval for the population mean lifespan of a basketball shoe is,

The 99.7% confidence interval for the population mean lifespan of a basketball shoe is (26.6436 months, 27.3564 months).

c)