Answer to Question #278947 in Statistics and Probability for nickey

Question

The manager of a restaurant in a large city claims that waiters working in all restaurants in his city earn an average of $150 or more in tips per week. A random sample of 25 waiters selected from restaurants of this city yielded a mean of $139 in tips per week with a standard deviation of $28. Assume that the weekly tips for all waiters in this city have a normal distribution.

 

a. Using the 1% significance level, can you conclude that the manager’s claim is true? Use

    both approaches. (20 Mks)

 

b. What is the Type I error in this exercise? Explain. What is the probability of making such  

    an error? (5 Mks)

 

 

Answer 1

a.

$H_0:\mu=150$ , waiters working in all restaurants earn an average of $150 or more in tips per week

$H_a:\mu<150$ , waiters working in all restaurants earn an average of less than $150 in tips per week

$t=\frac{\overline{x}-\mu}{\sigma/\sqrt n}=\frac{139-150}{28/5}=-1.964$

$df=n-1=27$

critical value:

$t_{crit}=2.492$

Since $|t|<t_{crit}$ we accept null hypothesis: waiters working in all restaurants earn an average of $150 or more in tips per week.

b.

a Type I error means rejecting the null hypothesis when it's actually true

it can happen if null hypothesis is true, but we get $|t|>t_{crit}$

The probability of making a type I error is α, which is the level of significance, in our case

this is 1%