Question

1. A scientist wishes to estimate the average depth of a river. He wants to be 95% confident that the estimate is accurate within 3 feet. From a previous study, the standard deviation of the depths measured was 5.48 feet. How large a sample is required?

1. The lifetime of a certain brand of battery is known to have a standard deviation of 23 hours. Suppose that a random sample of 100 such batteries has a mean lifetime of 39.1 hours. Based on this sample, find a 90% confidence interval for the true mean lifetime of all batteries of this brand. Then complete the table below. Carry your intermediate computations to at least three decimal places. Round your answers to one decimal place
   1. a) What is the lower limit of the 90% confidence interval?
   2. .What is the upper limit of the 90% confidence interval?

Answer 1

Solution :

Given that,

1) Population standard deviation = = 5.48

Margin of error = E = 3

At 95% confidence level the z is,

= 1 - 95%

= 1 - 0.95 = 0.05

/2 = 0.025

Z/2 = Z0.025 = 1.96  

sample size = n = [Z/2* / E] 2

n = [1.96 * 5.48 / 3]2

n = 12.81

Sample size = n = 13

2) Given that,

Point estimate = sample mean = = 39.1

Population standard deviation =    = 23

Sample size = n = 100

At 90% confidence level

= 1 - 90%  

= 1 - 0.90 =0.10

/2 = 0.05

Z/2 = Z0.05 = 1.645

Margin of error = E = Z/2 * ( /n)

= 1.645 * ( 23 /  100 )

= 3.8

At 90% confidence interval estimate of the population mean is,

  ± E

39.1 ± 3.8

( 35.3, 42.9 )  

lower limit = 35.3

upper limit = 42.9