In a recent sample of 84 used car sales costs, the sample mean was $6,425 with a

Question

In a recent sample of 84 used car sales costs, the sample mean was $6,425 with a standard deviation of $3,156. Assume the underlying distribution is approximately normal.
a. Which distribution should you use for this problem? Explain your choice.
b. Define the random variable X¯ in words.
c. Construct a 95% confidence interval for the population mean cost of a used car.
i. State the confidence interval.
ii. Calculate the error bound.
d. Explain what a “95% confidence interval” means for this study.

Explanation / Answer

Standard Error= sd/ Sqrt(n)
Where,
sd = Standard Deviation
n = Sample Size
Standard deviation( sd )=3156
Sample Size(n)=84
Standard Error = ( 3156/ Sqrt ( 84) )
= 344.348

Confidence Interval
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)=6452
Standard deviation( sd )=3156
Sample Size(n)=84
Confidence Interval = [ 6452 ± t a/2 ( 3156/ Sqrt ( 84) ) ]
= [ 6452 - 1.989 * (344.348) , 6452 + 1.989 * (344.348) ]
= [ 5767.092,7136.908 ]

Interpretations:
1) We are 95% sure that the interval [5767.092 , 7136.908 ] contains the true population mean
2) If a large number of samples are collected, and a confidence interval is created
for each sample, 95% of these intervals will contains the true population mean  

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