Consider the following random sample of diameter measurements (in inches) of 20 softballs. 4.69,4.83,4.89,4.77,4.81,4.71,4.87,4.77,4.71,4.7,4.72,4.83,4.86,4.85,4.82,4.72,4.85,4.75,4.7,4.86. If we assume that the diameter measurements are normally distributed, find a 99% confidence interval for the mean diameter of a softball. Give the lower limit and upper limit of the 99% confidence interval.

Carry your intermediate computations to at least three decimal places. Round your answers to two decimal places.

To find the 99% confidence interval for the mean diameter of the softballs, we'll follow these steps:

### Step 1: Identify the given information
- Sample size (\(n\)): 20
- Confidence level: 99%
- The diameter measurements: \(4.69, 4.83, 4.89, 4.77, 4.81, 4.71, 4.87, 4.77, 4.71, 4.70, 4.72, 4.83, 4.86, 4.85, 4.82, 4.72, 4.85, 4.75, 4.70, 4.86\)

### Step 2: Calculate the sample mean (\(\bar{x}\)) and sample standard deviation (\(s\))
We'll first compute the sample mean and standard deviation.

The sample mean (\(\bar{x}\)) is calculated as:
\[
\bar{x} = \frac{\sum x_i}{n}
\]

The sample standard deviation (\(s\)) is calculated using:
\[
s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}
\]

Let's compute these values.

### Step 3: Find the t-score for a 99% confidence level
Since the sample size is 20, we'll use the t-distribution. The degrees of freedom (\(df\)) is \(n-1 = 19\).

For a 99% confidence level and 19 degrees of freedom, the t-score (\(t_{\alpha/2}\)) can be found from the t-distribution table or using a calculator. The t-score is approximately 2.861.

### Step 4: Calculate the standard error (SE)
The standard error of the mean is given by:
\[
\text{SE} = \frac{s}{\sqrt{n}}
\]

### Step 5: Compute the margin of error (ME)
The margin of error is given by:
\[
\text{ME} = t_{\alpha/2} \times \text{SE}
\]

### Step 6: Determine the confidence interval
The confidence interval is calculated as:
\[
\text{Confidence Interval} = \bar{x} \pm \text{ME}
\]

Now, I'll compute the necessary intermediate values: sample mean, sample standard deviation, standard error, and margin of error.Here are the results based on the calculations:

- **Sample mean** (\(\bar{x}\)): 4.7855 inches
- **Sample standard deviation** (\(s\)): 0.0683 inches
- **Standard error** (\(\text{SE}\)): 0.0153 inches
- **Margin of error** (\(\text{ME}\)): 0.0437 inches

### Final Answer:
The 99% confidence interval for the mean diameter of the softballs is:
- **Lower limit**: 4.74 inches
- **Upper limit**: 4.83 inches

This means that we are 99% confident that the true mean diameter of the softballs lies between 4.74 inches and 4.83 inches.

Would you like any further details or have any questions?

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Here are some related questions for further exploration:

1. How would the confidence interval change if the confidence level was 95%?
2. What impact would a larger sample size have on the confidence interval?
3. What if the standard deviation of the population was known instead of the sample?
4. How would the confidence interval change if the sample mean were slightly higher?
5. What does a wider confidence interval indicate about the data?
6. How do you interpret the t-score in this context?
7. What would happen if the sample size was smaller (e.g., 10 softballs)?
8. Can we still use the t-distribution if the data is not normally distributed?

**Tip:** Always remember that increasing the sample size can lead to a more precise confidence interval by reducing the standard error.

Learn how to calculate a 99% confidence interval for the mean diameter of softballs with a step-by-step solution. Includes sample mean calculation, sample standard deviation computation, and margin of error determination. Suitable for advanced high school or college students.

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