Question

Before collecting any sample data, the fast-food chain in Problem below stipulates that the level of Type I error should be no larger than α = 0:01, and that if the number of cars passing the site is at or above µ= 210 per hour, then the level of Type II error also should not exceed β= 0:01.

The population standard deviation is estimated as being no larger than σ= 40. What sample size is required to achieve these objectives?
Ans. n= 347:45 = 348.

Base problem

A fast-food chain will build a new outlet in a proposed location only if more than 200 cars per hour pass the location during certain hours. For 20 randomly sampled hours during the designated hours, the average number of cars passing the location is X= 208:5 with s= 30:0. The statistical population is assumed to be approximately normal. The management of the chain conservatively adopted the alternative hypothesis H1:µ > 200:0.

Thanks in advance for your help by solving this doubt!

Answer 1

For question 1 the required formula for sample size is

ES= 210-200/40

ES= 0.25

For 99% power Z0.99 =2.33

n= (4.66/0.25)^2

n= (18.64)^2

n= 347.449

n= 348

2) NULL HYPOTHESIS H0:

ALTERNATIVE HYPOTHESIS Ha:

t= 208.5-200/30/sqrt(20)

t= 8.5/30/4.47

t= 8.5/6.71

t= 1.27

degrees of freedom= n-1=20-1=19

t critical= 1.73

Since t critical is greater than tcal therefore we are failed to reject the null hypothesis H0.

Conclusion: We don't have sufficient evidence to show that fast-food chain will build a new outlet in a proposed location only if more than 200 cars per hour pass the location during certain hours.