Question 1184610: The diameter of a brand of tennis balls is approximately normally distributed, with a mean of 2.67 inches and a standard deviation of 0.05 inch. A random sample of 10 tennis balls is selected. Complete parts (a) through (d) below.
The probability is 60% that the sample mean will be between what two values symmetrically distributed around the population mean?
The lower bound is---- inches
The upper bound is----- inches.
(Round to two decimal places as needed.)
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Here's how to calculate the bounds for the sample mean:
1. **Find the z-score corresponding to the desired probability:**
Since we want the probability that the sample mean falls *between* two values symmetrically distributed around the population mean, and that probability is 60%, we need to find the z-scores that cut off the *remaining* 40% (100% - 60% = 40%). Since the distribution is symmetric, each tail will have 20% (40% / 2 = 20%).
Look up the z-score corresponding to 0.20 (or 20%) in a standard normal (z) table. Since we are looking for the middle 60%, we need the z-score that corresponds to 0.70 (or 70%) or 1 - 0.20 on the left side of the distribution. The z-score is approximately 0.52. (You can also use a calculator or statistical software to find this z-score more precisely, but 0.52 is generally sufficient for this type of problem.)
2. **Calculate the standard error of the mean:**
The standard error of the mean (SEM) is the standard deviation of the sample means. It's calculated as:
SEM = σ / √n
where σ is the population standard deviation (0.05 inch) and n is the sample size (10).
SEM = 0.05 / √10 ≈ 0.0158 inch
3. **Calculate the margin of error:**
The margin of error is how much the sample mean is likely to vary from the population mean. It's calculated as:
Margin of Error = z * SEM
Margin of Error = 0.52 * 0.0158 ≈ 0.0082 inch
4. **Calculate the lower and upper bounds:**
* **Lower Bound:** Population Mean - Margin of Error = 2.67 - 0.0082 ≈ 2.66 inches
* **Upper Bound:** Population Mean + Margin of Error = 2.67 + 0.0082 ≈ 2.68 inches
**Therefore:**
The lower bound is approximately 2.66 inches.
The upper bound is approximately 2.68 inches.
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