Archeologists have studied sizes of Egyptian skulls in an attempt to determine whether breeding occurred between different cultures. Listed below are the widths​ (mm) of skulls from 150 A.D. Construct a 99​% confidence interval estimate of the mean skull width.  
127.9
137.6
126.4
132.3
143.2
135.1
138.9
128.6
 
Question content area bottom
Part 1
  
enter your response here mmless thanmuless than
  
enter your response here mm
​(Round to two decimal places as​ needed.)

To construct a 99% confidence interval estimate for the mean skull width, we need to follow these steps:

1. **Calculate the sample mean (\(\bar{x}\))**: The average of the given skull widths.
2. **Calculate the sample standard deviation (s)**: The standard deviation of the skull widths.
3. **Determine the appropriate t-score**: Since the sample size is small (n < 30), we'll use the t-distribution. The degrees of freedom (df) will be \(n - 1\).
4. **Calculate the margin of error (E)**: This is the product of the t-score and the standard error of the mean.
5. **Construct the confidence interval**: The confidence interval is given by \(\bar{x} \pm E\).

Let's go through the calculations:

### Step 1: Calculate the sample mean (\(\bar{x}\))

\[
\bar{x} = \frac{\sum x_i}{n}
\]

\[
\bar{x} = \frac{127.9 + 137.6 + 126.4 + 132.3 + 143.2 + 135.1 + 138.9 + 128.6}{8}
\]

\[
\bar{x} \approx 133.75 \text{ mm}
\]

### Step 2: Calculate the sample standard deviation (s)

\[
s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}
\]

Let's calculate the variance and then the standard deviation.

### Step 3: Determine the t-score

For a 99% confidence level and 7 degrees of freedom (df = 8 - 1), the t-score can be found using a t-distribution table or calculator.

### Step 4: Calculate the Margin of Error (E)

\[
E = t \times \frac{s}{\sqrt{n}}
\]

### Step 5: Construct the Confidence Interval

\[
\text{Confidence Interval} = \left(\bar{x} - E, \bar{x} + E\right)
\]

I'll compute this now.The calculations yield the following results:

- **Sample Mean (\(\bar{x}\))**: \(133.75\) mm
- **Sample Standard Deviation (s)**: \(5.97\) mm (rounded to two decimal places)
- **t-score for 99% confidence level**: \(3.50\) (rounded to two decimal places)
- **Margin of Error (E)**: Calculated as part of the confidence interval

### 99% Confidence Interval for the Mean Skull Width:
\[
(126.37 \text{ mm}, 141.13 \text{ mm})
\]

This means we are 99% confident that the true mean skull width of the population lies between 126.37 mm and 141.13 mm.

Would you like more details on any part of this process or have any other questions?

Here are 5 related questions:
1. How would the confidence interval change if we used a 95% confidence level instead?
2. What is the effect of increasing the sample size on the confidence interval?
3. How does the standard deviation influence the width of the confidence interval?
4. What is the significance of using the t-distribution instead of the normal distribution in this case?
5. Can you calculate the confidence interval if you only had the mean and standard deviation?

**Tip:** Always round your final confidence interval values to a suitable number of decimal places, usually to match the precision of the original data.

Archeologists have studied sizes of Egyptian skulls in an attempt to determine whether breeding occurred between different cultures. Listed below are the widths (mm) of skulls from 150 A.D. Construct a 99% confidence interval estimate of the mean skull width: 127.9, 137.6, 126.4, 132.3, 143.2, 135.1, 138.9, 128.6.

Calculate a 99% confidence interval for the mean width of Egyptian skulls from 150 A.D. using descriptive statistics, including sample mean, standard deviation, and margin of error. Learn key concepts in statistics, such as confidence intervals and t-distribution, with detailed step-by-step explanations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation