Answer:
Step-by-step explanation:
We would construct a confidence interval.
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.
Since the sample size is large and the population standard deviation is known, we would use the following formula and determine the z score from the normal distribution table.
Margin of error = z × σ/√n
Where
σ = population standard Deviation
n = number of samples
From the information given
x = 36.02 inches
σ = 0.1 inches
n = 40
To determine the z score, we subtract the confidence level from 100% to get α
α = 1 - 0.99 = 0.01
α/2 = 0.01/2 = 0.005
This is the area in each tail. Since we want the area in the middle, it becomes
1 - 0.005 = 0.995
The z score corresponding to the area on the z table is 2.58. Thus, confidence level of 99% is 2.58
Margin of error = 2.58 × 0.1/√40 = 0.041
Confidence interval = 36.02 ± 0.041
The lower end of the confidence interval is
36.02 - 0.041 = 35.979 inches
The upper end of the confidence interval is
36.02 - 0.041 = 36.061 inches
Since 36.061 > 36.05, then we would conclude that the machine needs an adjustment.
The formula used is
Confidence interval =
Sample mean ± z × σ/√n
