Question

The​ quality-control manager at a compact fluorescent light bulb​ (CFL) factory needs to determine whether the mean life of a large shipment of CFLs is equal to 7,530 hours. The population standard deviation is 92 hours. A random sample of 64 light bulbs indicates a sample mean life of 7,507 hours.

a. At the 0.05 level of​ significance, is there evidence that the mean life is different from 7,530 hours?

   What is/are the critical values?

b. Compute the​ p-value and interpret its meaning.

c. Construct a 95​% confidence interval estimate of the population mean life of the light bulbs.

d. Compare the results of​ (a) and​ (c). What conclusions do you​ reach?

Answer 1

Given :-

Population Standard Deviaiton = 92

Sample Mean = 7507

Sample Size n = 64

To Test :-

H₀ :- The mean life of a large shipment of CFLs is equal to 7,530 hours i.e  

H₁ :- The mean life of a large shipment of CFLs is different than 7,530 hours i.e  

Test Statistic :-

Z = -2

Test Criteria :-

Reject null hypothesis if |Z| > Z_/2

Z_/2 = Z_0.05/2 = Z_0.025 = 1.96 ( Critical Value)

|Z| > Z_/2 = 2 > 1.96, we reject the null hypothesis

Conclusion :- Accept Alternative Hypothesis

The mean life of a large shipment of CFLs is different than 7,530 hours i.e  ​

Part b) P value

P ( Z < -2 ) = 0.02275  

Since Alternative hypothesis is of two sided ( two tail test )

P value = 0.02275 * 2 = 0.0455

Conclusion based on P value

Reject null hypothesis if P value < ( level of significance)

0.0455 < 0.05, hence we reject null hyothesis

A smaller the p value stronger the evidence against the null hypothesis, so we reject the null hyothesis

larger the P value weaker the evidence against the null hypothesis, so we fail to reject null hypothesis

Part c) Construct a 95​% confidence interval estimate of the population mean life of the light bulbs.

Lower Limit =

Lower Limit = = 7484.46

Upper Limit =

Upper Limit = = 7529.54

Part d)

the mean life of a large shipment of CFLs is equal to 7,530 hours

But 7530 does not lie in the confidence interval (7484.46 , 7529.54)

Hence the average life of a large shipment of CFL is different than 7530 hours