The Ledd Pipe Company has received a large shipment of pipes, and a quality control inspector wishes to
estimate the average diameter of these pipes. A random sample of 18 pipes produces an average diameter of
2.56 mm with a standard deviation of 0.05 mm. You may assume that the diameters of the pipes are normally
distributed.
(A) Give a 95% confidence interval for the average diameter of pipes in this shipment.
95% = mean ± 1.695 SEm
SEm = SD/√n
To calculate a confidence interval for the average diameter of the pipes in this shipment, we can use the formula:
Confidence Interval = Sample Mean ± (Critical Value x Standard Error)
Step 1: Calculate the standard error.
The standard error (SE) represents the standard deviation of the sampling distribution. It can be calculated using the formula:
SE = Standard Deviation / √(Sample Size)
In this case, the standard deviation (σ) is given as 0.05 mm, and the sample size (n) is 18. Plugging in these values:
SE = 0.05 / √(18) = 0.05 / 4.24264 ≈ 0.01176 mm
Step 2: Find the critical value.
The critical value corresponds to the desired level of confidence and the degrees of freedom. With a sample size of 18, the degrees of freedom is n-1 = 18-1 = 17.
For a 95% confidence level, the critical value can be found using a t-distribution table or a statistical calculator. For simplicity, let's assume a two-tailed test, where we divide the desired confidence level by 2 to find the area in each tail.
Look up the critical value for α/2 = 0.05/2 = 0.025, with 17 degrees of freedom. This value is approximately 2.110.
Step 3: Calculate the confidence interval.
Using the formula for the confidence interval:
Confidence Interval = Sample Mean ± (Critical Value x Standard Error)
Plugging in the values:
Confidence Interval = 2.56 ± (2.110 x 0.01176) ≈ 2.56 ± 0.0248
The 95% confidence interval for the average diameter of the pipes in this shipment is approximately (2.5352 mm, 2.5848 mm).