In this chapter confidence intervals are introduced There w

: In this chapter, confidence intervals are introduced. There were 3 different methods of constructing a confidence interval, using a z-value, a t-value and with a proportion. There were also 2 methods of determining how large a sample size would be required to construct a confidence interval with the given parameters. One was done using the probability and the other using the given proportion. Below there are 4 confidence interval problems which will cover all three methods and 2 sample size problems, one for each method.

A manufacturer of Maserati cars wishes to investigate the length of time a car will last while driving over 200 mph. The production of Maseratis is very expensive, thus the number available for testing is small. A sample of 8 cars had a mean driving time of 2.25 hours with a standard deviation of 0.5 hours.

a. What is the population mean? What is the best estimate of that value?

b. Construct a 95 percent confidence interval for the population mean.

c. Explain why the t distribution is used as a part of the confidence interval.

d. Is it reasonable for the manufacturer to claim that the cars will last 2.5 hours?

Solution

CI = x ± t a/2 * (sd/ Sqrt(n))
Where,
x = Mean
sd = Standard Deviation
a = 1 - (Confidence Level/100)
ta/2 = t-table value
CI = Confidence Interval
Mean(x)=2.25
Standard deviation( sd )=0.5
Sample Size(n)=8
Confidence Interval = [ 2.25 ± t a/2 ( 0.5/ Sqrt ( 8) ) ]
= [ 2.25 - 2.365 * (0.18) , 2.25 + 2.365 * (0.18) ]
= [ 1.83,2.67 ]

a. What is the population mean? What is the best estimate of that value?
best estimate = Sample Mean(x)=2.25

b. Construct a 95 percent confidence interval for the population mean.
[ 1.83,2.67 ]

c. Explain why the t distribution is used as a part of the confidence interval.
Population standard deviation is unknown, sample size is less than value of 30

d. Is it reasonable for the manufacturer to claim that the cars will last 2.5 hours?
Yes, value lies in the interval calculated in part b.