Question

The Whitcomb Company manufactures a metal ring for industrial engines that usually weighs about 50 ounces. A random sample of 50 of these metal rings produced the following wights (in ounces).

51	53	56	50	44	47
53	53	42	57	46	55
41	44	52	56	50	57
44	46	41	52	69	53
57	51	54	63	42	47
47	52	53	46	36	58
51	38	49	50	62	39
44	55	43	52	43	42
57	49

(a) Construct a 95% confidence interval for the average weight of the metal rings made by Whitcomb Company.

(b) Test at 5% significance level to determine if the metal rings made by Whitcomb Company have an average weight less than 50 ounces.

(Hint: you have to first calculate the sample size, sample mean, and sample standard deviation; and then apply the appropriate statistical inference techniques)

Answer 1

a)

Level of Significance ,    α =    0.05          
degree of freedom=   DF=n-1=   49          
't value='   tα/2=   2.0096   [Excel formula =t.inv(α/2,df) ]      
                  
Standard Error , SE = s/√n =   6.8044   / √   50   =   0.962293
margin of error , E=t*SE =   2.0096   *   0.96229   =   1.933801
                  
confidence interval is                   
Interval Lower Limit = x̅ - E =    49.84   -   1.933801   =   47.906199
Interval Upper Limit = x̅ + E =    49.84   -   1.933801   =   51.773801
95%   confidence interval is (   47.91   < µ <   51.77   )

b)

Ho :   µ =   50                  
Ha :   µ <   50       (Left tail test)          
                          
Level of Significance ,    α =    0.05                  
sample std dev ,    s = √(Σ(X- x̅ )²/(n-1) ) =   6.8044                  
Sample Size ,   n =    50                  
Sample Mean,    x̅ = ΣX/n =    49.8400                  
                          
degree of freedom=   DF=n-1=   49                  
                          
Standard Error , SE = s/√n =   6.8044   / √    50   =   0.9623      
t-test statistic= (x̅ - µ )/SE = (   49.840   -   50   ) /    0.9623   =   -0.17
                             
                          
p-Value   =   0.4343   [Excel formula =t.dist(t-stat,df) ]              
Decision:   p-value>α, Do not reject null hypothesis

                      
Conclusion: There is not enough evidence that metal rings made by Whitcomb Company have an average weight less than 50 ounces

  

Thanks in advance!

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