The diameter of small Nerf balls manufactured at a factory in China is expected to be approximately normally distributed with a mean of 5.2 inches and a standard deviation of .08 inch. Suppose a random sample of 20 balls is selected. Find the interval that contains 87% of the sample means.
87% = mean ± 1.51 SEm
SEm = SD/√n
What is square root n?
The diameter of small Nerf balls manufactured at a factory in China is expected to be approximately normally distributed with a mean of 5.2 inches and a standard deviation of .08 inch. Suppose a random sample of 20 balls is selected. Find the interval that contains 87% of the sample means
To find the interval that contains 87% of the sample means, we need to calculate the confidence interval.
1. Find the sample mean:
μ = 5.2 inches
2. Find the standard deviation of the sample mean (also known as the standard error):
σ/√n = 0.08 inch / √20 ≈ 0.01787 inch
3. Determine the critical value for a given level of confidence. Since we want to find the interval that contains 87% of the sample means, we need to find the z-value that corresponds to an 87% central area in the standard normal distribution.
The central area (in both tails) corresponds to (100% - 87%) / 2 = 6.5% in each tail. Dividing this by 2 gives 3.25% in each tail. We can use a standard normal distribution table or a statistical calculator to find the z-value corresponding to a cumulative probability of 1 - 3.25% = 96.75%.
Using the cumulative distribution function (CDF) of the standard normal distribution, we find that the corresponding z-value is approximately 1.15.
4. Calculate the confidence interval:
Lower limit: μ - z * (σ/√n) = 5.2 - 1.15 * 0.01787 ≈ 5.178 inch
Upper limit: μ + z * (σ/√n) = 5.2 + 1.15 * 0.01787 ≈ 5.222 inch
Therefore, the interval that contains 87% of the sample means is approximately (5.178 inch, 5.222 inch).