Question

he owner of a local nightclub has recently surveyed a random sample of n = 250 customers of the club. She would now like to determine whether or not the mean age of her customers is over 30. If so, she plans to alter the entertainment to appeal to an older crowd. If not, no entertainment changes will be made. Using a level of significance of 1%, what rejection region should she use?

		Reject H0 if t > 2.34.
		Reject H0 if t > 2.58.
		Reject H0 if t < -2.34.
		Reject H0 if t < -2.55.

Answer 1

Nightclub has recently surveyed a random sample of n = 250 customers of the club

Now sample size n = 250

Given level of significance of 1% , i.e $\alpha$ = 0.01 .

We wish to to determine whether or not the mean age of her customers is over 30 .

Hypoyhesis to test would be:-

H0 : u $\leq$ 30     { mean age of her customers may not be over 30 }

H1 : u > 30       { mean age of her customers is over 30 i.e significantly more than 30 }

Since alternative hypothesis is of " > " type , so this is one tail test , as we are testing weather mean age of her customers is over 30 or not .

So , Rejection region is given by $t_{\alpha }$

Where $t_{\alpha }$ is t-distributed with n-1 = 250-1 = 249 dgree of freedom at $\alpha$ = 0.01

Required t-critcal region or t-critical value can be obtained from statistical book or more accurately from any software like R/Excel

From R

> qt(1-0.01,df=249)
[1] 2.341417

Hence t-critical value is $t_{\alpha }$ = 2.341417 $\approx$ 2.34

Now since alternative hypothesis is of " > " type , we reject null hypothesis if calculated test statistics value t is greater than t-critical value   $t_{\alpha }$ .

So we reject H0 , if t is greater than 2.34   i.e if t > 2.34

So correct option is

Reject H0 if t > 2.34.