A sample of the length in inches for newborns is given below. Assume that lengths are normally distributed. Find the 99% confidence interval of the mean length.

Length
18.1
17.6
20.6
19.6
16.3
21.9
17.2
21
16.8
22.8


To find the 99% confidence interval for the mean length of newborns based on the provided sample, we will follow these steps:

### 1. Calculate the sample mean (\( \bar{x} \)):
We first need to find the mean of the given sample.

Given lengths:
18.1, 17.6, 20.6, 19.6, 16.3, 21.9, 17.2, 21, 16.8, 22.8

### 2. Calculate the sample standard deviation (\( s \)).

### 3. Use the t-distribution for the 99% confidence interval:
Since the sample size is small (\( n = 10 \)), we use the t-distribution. The formula for the confidence interval is:

\[
CI = \bar{x} \pm t_{\alpha/2} \times \frac{s}{\sqrt{n}}
\]

Where:
- \( \bar{x} \) is the sample mean
- \( t_{\alpha/2} \) is the t-critical value for a 99% confidence level and degrees of freedom \( df = n - 1 \)
- \( s \) is the sample standard deviation
- \( n \) is the sample size

I'll now calculate the necessary values for this.### Calculations:
- **Sample Mean (\( \bar{x} \))**: 19.19 inches
- **Sample Standard Deviation (\( s \))**: 2.30 inches
- **t-Critical Value for 99% Confidence Level**: 3.25

### 99% Confidence Interval:
The 99% confidence interval for the mean length is:

\[
(16.83 \text{ inches}, 21.55 \text{ inches})
\]

This means we are 99% confident that the true mean length of newborns falls between 16.83 and 21.55 inches.

Would you like more details or have any questions?

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**Relative Questions**:
1. How would the confidence interval change with a 95% confidence level?
2. What happens to the confidence interval if the sample size increases?
3. How does the confidence interval change if the variability (standard deviation) increases?
4. Can the normal distribution assumption affect the confidence interval accuracy?
5. What is the difference between using a t-distribution and a z-distribution in confidence intervals?

**Tip**: Always ensure the sample is large enough to justify the use of the t-distribution for small datasets!

A sample of the length in inches for newborns is given below. Assume that lengths are normally distributed. Find the 99% confidence interval of the mean length.

Length:
18.1, 17.6, 20.6, 19.6, 16.3, 21.9, 17.2, 21, 16.8, 22.8

Find the 99% confidence interval for the mean length of newborns using a sample of 10 lengths. This solution covers the key statistical concepts of confidence intervals, the t-distribution, and how to calculate the sample mean and standard deviation. Suitable for high school and college-level statistics.

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