Question

The quality-control manager at a compact flourescent light bulb factory wants to test the claim that the mean life of a large shipment of cfls is equal to 6500 hours. the population standard deviation is 500 hours. a random sample of 50 cfls indicates a sample mean life of 6,700 hours and sample standard deviation is 600 hours.

Critical values, where P(Z > Z) - a and P(t >) - a 2016-1282 20.06-1645 20.02 -1.960 10.10. - 1.299 10.06.45 - 1677 foto - 2010 1.115 pt.
a. At the 0.05 level of significance, is there evidence that the mean life is different from 6,500 hours?
b. Compute the p value and interpret its meaning.
c. Construct a 95% confidence interval estimate of the population mean life of the CFLs.
d. Compare the results of (a) and (c). What conclusions do you reach?

Answer 1

a. At the 0.05 level of significance, there is evidence that the mean life is different from 6,500 hours.

b. The p value= ≈ 0.00480 for z- test which is less than 0.05 and H0 is rejected .

The p value= 0.006913 for t- test which is less than 0.05 and H0 is rejected for 49 degrees of freedom.

c. CI [6583.336 ,6816.336]

d. The range of CI [6583.336 ,6816.336] tells that the cfls having a different mean life lie in this range.

Explanation:

Population mean = u= 6500 hours.

Population standard deviation = σ=500 hours.

Sample size =n= 50

Sample mean =x`= 6,700 hours

Sample standard deviation=s= 600 hours.

Critical values, where P(Z > Z) =∝ and P(t >) =∝

Z(0.10)=1.282

Z(0.05)=1.645

Z(0.025)=1.960

t(0.01)(49)= 1.299

t(0.05)= 1.677

t(0.025,49)=2.010

Let the null and alternate hypotheses be

H0: u = 6500 against the claim Ha: u ≠ 6500

Applying Z test

Z= x`- u/ s/√n

z= 6700-6500/500/√50

Z= 200/70.7113

z= 2.82=2.82

Applying t test

t= x`- u /s/√n

t= 6700-6500/600/√50

t= 2.82

a. At the 0.05 level of significance, there is evidence that the mean life is different from 6,500 hours.

Yes we reject H0 for z- test as it falls in the critical region,at the 0.05 level of significance, z=2.82 > z∝=1.645

For t test we reject H0 as it falls in the critical region,at the 0.05 level of significance, t=2.82 > t∝=1.677 with n-1 = 50-1 = 49 degrees of freedom.

b. The p value= ≈ 0.00480 for z- test which is less than 0.05 and H0 is rejected .

The p value= 0.006913 for t- test which is less than 0.05 and H0 is rejected for 49 degrees of freedom.

c. The 95 % confidence interval of the population mean life is estimated by

x` ± z∝/2 (σ/√n )

6700± 1.645 (500/√50)

6700±116.336

6583.336 ,6816.336

d. The range of CI [6583.336 ,6816.336] tells that the cfls having a different mean life lie in this range.