Answer: (b) 15.07 to 19.93 miles
Step-by-step explanation:
Confidence interval is written in the form,
(Sample mean - margin of error, sample mean + margin of error)
The sample mean, x is the point estimate for the population mean.
Margin of error = z × s/√n
Where
s = sample standard deviation = 3.8
n = number of samples = 20
From the information given, the population standard deviation is unknown and the sample size is small, hence, we would use the t distribution to find the z score
In order to use the t distribution, we would determine the degree of freedom, df for the sample.
df = n - 1 = 20 - 1 = 19
Since confidence level = 99% = 0.99, α = 1 - CL = 1 – 0.99 = 0.01
α/2 = 0.02/2 = 0.005
the area to the right of z0.005 is 0.025 and the area to the left of z0.025 is 1 - 0.005 = 0.995
Looking at the t distribution table,
z = 2.861
Margin of error = 2.861 × 3.8/√20
= 2.43
the lower limit of this confidence interval is
17.5 - 2.43 = 15.07 miles
the upper limit of this confidence interval is
17.5 + 2.43 = 19.93 miles
