The owner of a local nightclub has recently surveyed a random sample of n = 250

Question

The 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.

a) State the appropriate null hypothesis and alternative hypothesis. Explain how you obtain your answer.

b) If she wants to be 99% confident in her decision, what rejection region should she use? Explain how you obtain your answer.

c) Suppose she found that the sample mean was 30.45 years and the sample standard deviation was 5 years. If she wants to be 99% confident in her decision, what decision should she make? Explain how you obtain your answer.

d) Suppose she found that the sample mean was 30.45 years and the sample standard deviation was 5 years. If she wants to be 99% confident in her decision, what conclusion can she make? Explain how you obtain your answer.

e) Suppose she found that the sample mean was 30.45 years and the sample standard deviation was 5 years. What is the p-value associated with the test statistic? Explain how you obtain your answer.

Explanation / Answer

a)

Formulating the null and alternative hypotheses,              
              
Ho:   u   <=   30  
Ha:    u   >   30   [ANSWER]

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b)
              
              
Thus, getting the critical z, as alpha =    0.01   ,      
alpha = 1 - 0.99 =    0.01          

By table/technology,

zcrit =    +   2.326347874      

So we reject Ho when z > 2.326. [ANSWER]

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c)
              
Getting the test statistic, as              
              
X = sample mean =    30.45          
uo = hypothesized mean =    30          
n = sample size =    250          
s = standard deviation =    5          
              
Thus, z = (X - uo) * sqrt(n) / s =    1.423024947  
As z < 2.326, we fail to reject Ho. [ANSWER]

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d)      

There is no significant evidence that the true mean age of her customers is over 30. [CONCLUSION]

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E)
              
Also, the p value is, as this is right tailed,              
              
P(z>1.42302) = Pvalue =    0.077364462   [ANSWER]      
              

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