A machine produces parts with lengths that are normally distributed with ó = 0.52. A sample of 16 parts has a mean length of 75.07.
(a) Give a point estimate for ì. (Give your answer correct to two decimal places.)
75.07.
(b) Find the 99% confidence maximum error of estimate for ì. (Give your answer correct to three decimal places.0.33 .
(c) Find the 99% confidence interval for ì. (Give your answer correct to three decimal places.)
Lower Limit 74.686 .
Upper Limit 75.453 .
(a) The point estimate for the mean length, μ, of the parts is simply the sample mean, which is given as 75.07.
(b) To find the 99% confidence maximum error of estimate, we first need to find the critical value corresponding to a 99% confidence interval. Since the sample size is small (n = 16) and the population standard deviation (σ) is known, we can use a normal distribution.
The critical value can be found using a standard normal distribution table or a calculator. For a 99% confidence level, the critical value is approximately 2.617.
Now, the formula for the maximum error of estimate is:
Maximum error = critical value * standard deviation / square root of sample size
Here, the standard deviation (σ) is given as 0.52 and the sample size (n) is 16.
Maximum error = 2.617 * 0.52 / √16 ≈ 0.331
So, the 99% confidence maximum error of estimate for μ is approximately 0.331.
(c) To find the 99% confidence interval for μ, we use the formula:
Confidence interval = sample mean ± (critical value * standard deviation / square root of sample size)
Substituting the given values:
Confidence interval = 75.07 ± (2.617 * 0.52 / √16)
Confidence interval = 75.07 ± 2.617 * 0.13
Confidence interval = 75.07 ± 0.34
Lower Limit = 75.07 - 0.34 ≈ 74.686
Upper Limit = 75.07 + 0.34 ≈ 75.453
Therefore, the 99% confidence interval for μ is approximately 74.686 to 75.453.
To find the answers to these questions, you need to use the concepts of point estimate, maximum error of estimate, and confidence interval.
(a) Point Estimate for μ:
The point estimate for μ is the sample mean, which is given as 75.07. So the point estimate for μ is 75.07.
(b) Maximum Error of Estimate for μ:
The maximum error of estimate for μ can be calculated using the formula:
Maximum Error = (Z-score) * (Standard Deviation of the sample mean)
Since the population standard deviation (σ) is given as 0.52 and the sample size is 16, the standard deviation of the sample mean (σ̄) can be calculated as:
σ̄ = σ / √(sample size)
= 0.52 / √(16)
= 0.52 / 4
= 0.13
The Z-score for a 99% confidence level can be found using a standard normal distribution table. For a 99% confidence level, the Z-score is approximately 2.576.
Maximum Error = 2.576 * 0.13
= 0.335 (rounded to three decimal places)
Therefore, the 99% confidence maximum error of estimate for μ is 0.335.
(c) Confidence Interval for μ:
The confidence interval for μ can be calculated using the formula:
Confidence Interval = (Sample Mean) ± (Z-score) * (Standard Deviation of the sample mean)
Substituting the given values:
Confidence Interval = 75.07 ± 2.576 * 0.13
= 75.07 ± 0.335
= (74.735, 75.405) (rounded to three decimal places)
Therefore, the 99% confidence interval for μ is (74.735, 75.405).