Question

A camera film is currently manufactured to a thickness of 25 thousandths. The process engineer wants to increase the speed of the film and believes that this can be achieved by decreasing the film thickness to 20 thousandths. Film speed is measured in µJ / in2, which is a measure inversely proportional to speed, that is, at higher speed minus µJ / in2

8 sheets of each thickness are taken (25 mils and 20 mils) and the following results are obtained:

For 25 mils: xbar = 1.15 µJ / in2 and s = 0.11 µJ / in2

For 20 mils: xbar = 1.06 µJ / in2 and s = 0.09 µJ / in2

 a) Is there evidence to support the belief of the process engineer? (Assume that σ1 = σ2) (10 pts) -

Clearly define the hypothesis, the areas of acceptance and rejection, the test statistic and express your conclusion in complete sentence
b) Calculate the Pvalue for this test (5 pts)

c) Is the presumption of equal variances supported?

Answer 1

a)

Ho :   µ1 - µ2 =   0
Ha :   µ1-µ2 >   0
      
Sample #1   ---->   1
mean of sample 1,    x̅1=   1.150
standard deviation of sample 1,   s1 =    0.110
size of sample 1,    n1=   8
      
Sample #2   ---->   2
mean of sample 2,    x̅2=   1.060
standard deviation of sample 2,   s2 =    0.090
size of sample 2,    n2=   8
      
difference in sample means = x̅1 - x̅2 =    1.15-1.06=   0.0900
pooled std dev , Sp=   √([(n1 - 1)s1² + (n2 - 1)s2²]/(n1+n2-2)) =    0.1005
std error , SE =    Sp*√(1/n1+1/n2) =    0.0502
t-statistic = ((x̅1-x̅2)-µd)/SE =    (0.09-0)/0.0502=   1.7911
      
Level of Significance ,    α =    0.05
Degree of freedom, DF=   n1+n2-2 =    14
t-critical value , t* =   1.761   (excel function: =t.inv(α,df)
Decision:     t-stat > critical value , so, Reject Ho  

b)

p-value =    0.047460   [excel function: =T.DIST.RT(t stat,df) ]

c)

Ho : σ₁   =   σ₂
Ha : σ₁   ╪   σ₂
      
Sample Standard deviation,    s₁ =    0.11
Sample size,    n₁ =    8
Sample Standard deviation,    s₂ =    0.09
Sample size,    n₂ =    8
F test Statistic,F = s₁² / s₂² =   0.11²/0.09²=   1.494
      
Degree of freedom:      
df₁ = n₁-1 =        7
df₂ = n₂-1 =        7
P value=       0.6095
Decision:   Do not reject the null hypothesis  

so, assumption of equal variance is supported