Please explain the answer The life in hours of a 75-watt light bulb is known to

Question

Please explain the answer The life in hours of a 75-watt light bulb is known to be normally distributed with sigma = 25 hours. A random sample of 20 bulbs has a mean life of x = 1014 hours. a) Construct a 95% two-sided confidence interval on the mean life. b) Construct a 95% lower-confidence bound on the mean life. (c) Suppose in a) above (i.e. 95% two-sided CI) we want the error in estimating the mean life to be five hours. What sample size should be used?. d) Suppose we want the total width of the 95% two-sided CI to be six hours. What sample size should be used?

Explanation / Answer

Part-a

95% confidence interval =xbar±z0.05/2*/sqrt(n)

=1014-1.96*25/sqrt(20) , 1014+1.96*25/sqrt(20)

=(1003.04            1024.96)

Part-b

Upper tailed critical z0.05=1.645

Lower confidence bound= xbar+z0.05*/sqrt(n)

=1014+1.645*25/sqrt(20)

=1023.20

Part-c

Margin of error E=5

Critical z=1.96

The =25

So sample size n>= (z* /E)2=(1.96*25/5)2= 96.04

Hence, a sample of size at least 97 required

Part-d

Margin of error E=6/2=3

Critical z=1.96

The =25

So sample size n>= (z* /E)2=(1.96*25/3)2= 266.78

Hence, a sample of size at least 267 required

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