(3 points) The Highway Safety Department wants to study the driving habits of individuals. A sample of 49 cars traveling on a particular stretch of highway revealed an average speed of 68.7 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 0.5 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 =
Solution :
Given that,
1) Point estimate = sample mean = = 68.7
sample standard deviation = s = 9.8
sample size = n = 49
Degrees of freedom = df = n - 1 = 49 - 1 = 48
At 99% confidence level
= 1 - 99%
=1 - 0.99 =0.01
/2
= 0.005
t/2,df
= t0.005,48 = 2.682
Margin of error = E = t/2,df * (s /n)
= 2.682 * (9.8 / 49)
Margin of error = E =3.75
The 99% confidence interval estimate of the population mean is,
± E
= 68.7 ± 3.75
= ( 64.95, 72.45)
2) margin of error = E = 0.5
sample size = n = [t/2,df* s / E] 2
n = [2.682 * 9.8 / 0.5]2
n = 2763.31
Sample size = n = 2764
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