Question

The Highway Safety Department wants to study the driving habits of individuals. A sample of 34 cars traveling on a particular stretch of highway revealed an average speed of 68.8 miles per hour with a standard deviation of 9.8 miles per hour. Round to 4 decimal places.

1.Calculate a 99% confidence interval for the true mean speed of all cars on this particular stretch of highway.

( , )

2. What sample size is needed to estimate the true average speed to within 2.5 mph at 99% confidence? Note: For consistency's sake, round your t* value to 3 decimal places before calculating the necessary sample size.

Choose n =

Answer 1

Solution :

Given that,

Point estimate = sample mean = = 68.8

sample standard deviation = s = 9.8

sample size = n = 34

Degrees of freedom = df = n - 1 = 34 - 1 = 33

1) At 99% confidence level

= 1 - 99%

=1 - 0.99 =0.01

/2 = 0.005

t/2,df = t0.005,33 = 2.733

Margin of error = E = t/2,df * (s /n)

= 2.733 * (9.8 / 34 )

Margin of error = E = 4.5933

The 99% confidence interval estimate of the population mean is,

  ± E  

= 68.8 ± 4.5933

= ( 64.2067, 73.3933 )

2) margin of error = E = 2.5

sample size = n = [t/2,df* s / E]2

n = [2.733 * 9.8 / 2.5 ]2

n = 114.77

Sample size = n = 115