Question

The Highway Safety Department wants to study the driving habits of individuals. A sample of 40 cars traveling on a particular stretch of highway revealed an average speed of 69.9 miles per hour with a standard deviation of 5.8 miles per hour. Round to 4 decimal places.

1.Calculate a 95% 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 mph at 95% 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 = = 69.9

sample standard deviation = s = 5.8

sample size = n = 40

Degrees of freedom = df = n - 1 = 40 - 1 = 39

1) At 95% confidence level

= 1 - 95%

=1 - 0.95 =0.05

/2 = 0.025

t/2,df = t0.025,39 = 2.023

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

= 2.023 * (5.8 / 40)

Margin of error = E = 1.8552

The 95% confidence interval estimate of the population mean is,

  ± E  

= 69.9 ± 1.8552

= ( 68.0448, 71.7552 )

2) margin of error = E = 2

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

n = [2.023 * 5.8 / 2 ]2

n = 34.41

Sample size = n = 35