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 7.505 hours. The population standard deviation is 100 hours. A random sample of 64 light bulbs indicates a sample mean life of 7.480 hours.

a. At the 0.05 level of significance, is there evidence that the mean life is different from 7.505 hours?

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

(a) GIVEN:

Sample size of light bulbs

Sample mean life hours

Population standard deviation hours

HYPOTHESIS:

(That is, the true mean life is not significantly different from 7.505 hours)

(That is, the true mean life is significantly different from 7.505 hours)

LEVEL OF SIGNIFICANCE:

TEST STATISTIC:

which follows standard normal distribution

CALCULATION:

  

  

CRITICAL VALUE:

The two tailed z critical value at significance level is

DECISION RULE:

CONCLUSION:

Since the calculated z statistic (-0.002) is greater than the critical value (-1.96), we fail to reject the null hypothesis and conclude that the true mean life is not significantly different from 7.505 hours.

(b) P VALUE:

The p value is,

Using the z table, the probability value is the value with row value 0.0 and column 0.00

DECISION RULE:

CONCLUSION:

Since the calculated p value is greater than the significance level , we fail to reject the null hypothesis and conclude that the true mean life is not significantly different from 7.505 hours.

(c) 95% CONFIDENCE INTERVAL FOR POPULATION MEAN LIFE OF LIGHT BULB:

The 95% confidence interval for population mean is,

where is the z critical value at 95% confidence level is .

CALCULATION:

The 95% confidence interval for population mean life of light bulbs is,

  

  

The 95% confidence interval for population mean life of light bulbs is .

(d) CONCLUSION:

Since the value of the parameter specified by the null hypothesis is contained in the 95% interval , the null hypothesis cannot be rejected at the 0.05 level. Thus we fail to reject the null hypothesis and conclude that the true mean life is not significantly different from 7.505 hours.

Thus there is no sufficient evidence to prove that the mean life is different from 7.505 hours.