Question

a. The marketing manager for an automobile manufacturer is interested in determining the proportion of new compact-car owners who would have purchased a passenger-side inflatable air bag if it had been available for an additional cost of $300. The manager believes from previous information that the proportion is 0.30. Suppose that a survey of 200 new compact-car owners is selected and 79 indicate that they would have purchased the inflatable air bags. If you were to conduct a test to determine whether there is evidence that the proportion is different from 0.30 and decided not to reject the null hypothesis, what conclusion could you draw? Write your conclusion below. You have to just write your conclusion and not all 7-steps as per our 7-steps method.

b. The owner of a local nightclub has recently surveyed a random sample of n = 25 customers of the club. She would now like to determine whether or not the mean age of her customers is over 35. If so, she plans to alter the entertainment to appeal to an older crowd. If not, no entertainment changes will be made. Suppose she found that the sample mean was 38 years and the sample standard deviation was 3.5 years. What is the p-value associated with the test statistic? You need to show your calculations for p value below including any graphs that you need to draw.

c. A pharmaceutical company has come up with a new drug and has applied for FDA approval. To approve the drug, FDA must find that it is safe for patients. What would be a Type II error in this case? You need to write the statement for Type II error by QMB 500 first formulating a null and alternate hypothesis. (Hint: See a similar problem like this in practice questions for Chapter 9).

d. Suppose, in testing a hypothesis about a proportion, the Z test statistic is computed to be 2.04. What should be your decision about the null hypothesis if the chosen level of significance is 0.01 and a two-tailed test is used? Write your decision below and provide reason for your decision along with any relevant calculations to support your decision. Do not write all 7-steps from the 7-step method discussed in the class.

Answer 1

a)

Ho :   p =    0.3                  
H1 :   p ╪   0.3       (Two tail test)          
                          
Level of Significance,   α =    0.05                  
Number of Items of Interest,   x =   79                  
Sample Size,   n =    200                  
                          
Sample Proportion ,    p̂ = x/n =    0.3950                  
                          
Standard Error ,    SE = √( p(1-p)/n ) =    0.0324                  
Z Test Statistic = ( p̂-p)/SE = (   0.3950   -   0.3   ) /   0.0324   =   2.9318
                          
critical z value =    ±    1.960   [excel formula =NORMSINV(α/2)]              
                          
p-Value   =   0.003370431   [excel formula =2*NORMSDIST(z)]              
Decision:   p-value<α , reject null hypothesis               

there is evidence that the proportion is different from 0.30    

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b)

Ho :   µ =   35                  
Ha :   µ >   35       (Right tail test)          
                          
Level of Significance ,    α =    0.05                  
sample std dev ,    s =    3.5000                  
Sample Size ,   n =    25                  
Sample Mean,    x̅ =   38.0000                  
                          
degree of freedom=   DF=n-1=   24                  
                          
Standard Error , SE = s/√n =   3.5000   / √    25   =   0.7000      
t-test statistic= (x̅ - µ )/SE = (   38.000   -   35   ) /    0.7000   =   4.29
                             
                          
p-Value   =   0.0001   [Excel formula =t.dist(t-stat,df) ]              
Decision:   p-value<α, Reject null hypothesis

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d)

z stat = 2.04

critical z value, z* =      2.5758

z stat < z critical , do not reject Ho

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Please revert back in case of any doubt.

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