Question

If 36 measurements of the specific gravity of aluminum had a mean of 2.705 and a standard deviation of .028 , construct a \(98 \%\) confidence interval for the actual specific gravity of aluminum.

Short answer

Answer: The 98% confidence interval for the true specific gravity of aluminum based on the given sample is (2.69298, 2.71702).

Step-by-step solution

Identify the given information

Sample mean (x̅) = 2.705, Standard deviation (σ) = 0.028, Sample size (n) = 36, and Confidence level = 98%.

Find the critical value for 98% confidence interval

First, we need to find the critical value for the given confidence level. For a 98% confidence interval, there is a 2% chance that the true mean is not within the interval. Since we are using a two-tailed distribution (as we are estimating an upper and lower bound for the confidence interval), we need to find the critical value for the 99% confidence level (1 - (2% / 2)). Using a Z-table or calculator, we find that the critical value (Z) for a 99% confidence level is approximately 2.576.

Compute the standard error of the mean

The standard error of the mean is calculated using the formula: Standard Error (SE) = \(\frac{σ}{\sqrt{n}}\) Plug in the given values: SE = \(\frac{0.028}{\sqrt{36}}\) = \(\frac{0.028}{6}\) = 0.00466667

Calculate the margin of error

Margin of error (ME) is calculated using the formula: ME = Z * SE Plug in the values: ME = 2.576 * 0.00466667 ≈ 0.01202

Construct the 98% confidence interval

The confidence interval is calculated using the formula: Confidence Interval (CI) = (x̅ - ME, x̅ + ME) Plug in the values: CI = (2.705 - 0.01202, 2.705 + 0.01202) = (2.69298, 2.71702) Therefore, we can say with 98% confidence that the true specific gravity of aluminum lies within the interval (2.69298, 2.71702).