Question

At one point during the 2019 baseball season, Dodgers pitcher Hyun-Jin Ryu had thrown his 2-seam fastball 134 times. The mean speed of those pitches was 90.02 mph with a sample standard deviation of 1.32 mph.

Use this data to construct a 99% confidence interval for the mean speed of Ryu's 2-seam fastball.

______________ to _______________

The margin of error of this confidence interval is _______________

To obtain a margin of error of only 0.1 mph at the 99% confidence level, we would like to know how many 2-seam fastballs would need to be sampled. The critical value needed is _________

The sample size needed would be_____________

Answer 1

Solution :

Given that,

1) Point estimate = sample mean = = 90.02

sample standard deviation = s = 1.32

sample size = n = 134

Degrees of freedom = df = n - 1 = 134 - 1 = 133

t _/2,df = 2.61

Margin of error = E = t_/2,df * (s /n)

= 2.61 * ( 1.32/ 134)

Margin of error = E =0.298

The 99% confidence interval estimate of the population mean is,

- E < < + E

90.02 - 0.298 < < 90.02 + 0.298

89.722 < < 90.318

A 99% confidence interval for the mean speed of Ryu's 2-seam fastball.

89.722  to 90.318

The margin of error of this confidence interval is = E =0.298

2) Margin of error = E = 0.1

The critical value needed is Z_/2 = 2.58

sample size = n = [Z_/2* S / E] ²

n = [2.58 * 1.32/0.1 ]²

n = 1160

Sample size = n = 1160

The sample size needed would be n = 1160