Question

Number of defective monitors manufactured in day shift and afternoon shift is to be compared. A sample of the production from six day shifts and eight afternoon shifts revealed the following number of defects.

Day 4 5 8 6 7 9

Afternoon 9 8 10 7 6 14 11 5

Is there a difference in the mean number of defects per shift? Choose an appropriate significance level.

(a) State the null hypothesis and the alternative hypothesis.

(b) What is the decision rule?

(c) What is the value of the test statistic?

(d) What is your decision regarding the null hypothesis?

(e) What is the p-value?

(f ) Interpret the result.

(g) What assumptions are necessary for this test?

(Typed answer preferred)

Answer 1

a)

Ho :   µ1 - µ2 =   0
Ha :   µ1-µ2 ╪   0

b)

α=0.05

Degree of freedom, DF=   n1+n2-2 =    12  
t-critical value , t* = ± 2.179   (excel formula =t.inv(α/2,df)

reject Ho: if test stat <-2.179 or t > 2.179

c)

Sample #1   ---->   1                  
mean of sample 1,    x̅1=   6.500                  
standard deviation of sample 1,   s1 =    1.871                  
size of sample 1,    n1=   6                  
                          
Sample #2   ---->   2                  
mean of sample 2,    x̅2=   8.750                  
standard deviation of sample 2,   s2 =    2.915                  
size of sample 2,    n2=   8                  
                          
difference in sample means =    x̅1-x̅2 =    6.5000   -   8.8   =   -2.250  
                          
pooled std dev , Sp=   √([(n1 - 1)s1² + (n2 - 1)s2²]/(n1+n2-2)) =    2.5331                  
std error , SE =    Sp*√(1/n1+1/n2) =    1.3680                  
                          
t-statistic = ((x̅1-x̅2)-µd)/SE = (   -2.2500   -   0   ) /    1.37   =   -1.6447

d)

Decision:     t-stat >-2.179 , so, Do not Reject Ho  

e)

p-value =        0.1260   (excel function: =T.DIST.2T(t stat,df) )  

f)

There is not enough evidence that  there is a difference in the mean number of defects per shift

g)

sample are random and independent

population from which samples are taken should be normally distributed