Sherds of clay vessels were put together to reconstruct rim diameters of the original ceramic vessels at the Wind Mountain archaeological site†. Suppose that a random sample of ceramic vessels gave the following rim diameters (in centimeters).

16

13.5

22.2

12.8

13

19.7

11.8

13.4

17.6

18.2

(a)

Use a calculator with mean and sample standard deviation keys to find the sample mean *x* and sample standard deviation *s*. (Round your answers to four decimal places.)

*x*= cm*s*= cm

(b)

Compute an 80% confidence interval for the population mean 𝜇 of rim diameters for such ceramic vessels found at the Wind Mountain archaeological site. (Enter your answer in the form: lower limit to upper limit. Include the word "to." Round your numerical values to one decimal place.)
 
 cm

Let's break down the problem step by step.

### Part (a): Calculate the Sample Mean (\( \bar{x} \)) and Sample Standard Deviation (\( s \))

Given the rim diameters (in cm):
\[ 
16, \ 13.5, \ 22.2, \ 12.8, \ 13, \ 19.7, \ 11.8, \ 13.4, \ 17.6, \ 18.2 
\]

#### Step 1: Calculate the Sample Mean (\( \bar{x} \))

The sample mean is calculated by summing all the data points and then dividing by the number of data points.

\[
\bar{x} = \frac{16 + 13.5 + 22.2 + 12.8 + 13 + 19.7 + 11.8 + 13.4 + 17.6 + 18.2}{10}
\]

\[
\bar{x} = \frac{158.2}{10} = 15.82 \text{ cm}
\]

So, the sample mean \( \bar{x} \) is **15.8200 cm**.

#### Step 2: Calculate the Sample Standard Deviation (\( s \))

The sample standard deviation \( s \) is calculated using the formula:

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

Let's go ahead and compute the standard deviation.

The sum of squared deviations from the mean:

\[
\sum (x_i - \bar{x})^2 = (16 - 15.82)^2 + (13.5 - 15.82)^2 + \dots + (18.2 - 15.82)^2
\]

This calculation yields:

\[
s = \sqrt{\frac{91.856}{9}} \approx \sqrt{10.2062} \approx 3.1947 \text{ cm}
\]

So, the sample standard deviation \( s \) is **3.1947 cm**.

### Part (b): Compute an 80% Confidence Interval for the Population Mean \( \mu \)

For an 80% confidence interval, the corresponding z-score is approximately 1.2816 (from the standard normal distribution table).

#### Step 1: Calculate the Standard Error of the Mean (SEM)

The SEM is given by:

\[
\text{SEM} = \frac{s}{\sqrt{n}} = \frac{3.1947}{\sqrt{10}} \approx 1.0106 \text{ cm}
\]

#### Step 2: Calculate the Confidence Interval

The confidence interval is given by:

\[
\bar{x} \pm z \times \text{SEM} 
\]

Substituting the values:

\[
15.82 \pm 1.2816 \times 1.0106
\]

\[
15.82 \pm 1.2956 \text{ cm}
\]

The lower limit is:

\[
15.82 - 1.2956 \approx 14.5 \text{ cm}
\]

The upper limit is:

\[
15.82 + 1.2956 \approx 17.1 \text{ cm}
\]

So, the 80% confidence interval for the population mean \( \mu \) is **14.5 cm to 17.1 cm**.

### Final Answers:
- **(a) Sample Mean \( \bar{x} \)**: 15.8200 cm  
- **(a) Sample Standard Deviation \( s \)**: 3.1947 cm  
- **(b) 80% Confidence Interval for \( \mu \)**: 14.5 cm to 17.1 cm  

Would you like to delve deeper into any of these concepts or have any further questions?

---

Here are 8 related questions you might explore:

1. How does the choice of confidence level (e.g., 80% vs. 95%) affect the width of the confidence interval?
2. What would happen to the confidence interval if we had a larger sample size?
3. How do we interpret the standard deviation in the context of this archaeological data?
4. Why do we use the z-score for the confidence interval calculation, and how does it differ for t-distribution?
5. How would you calculate the confidence interval if the population standard deviation was known?
6. What assumptions are we making about the distribution of rim diameters in this confidence interval calculation?
7. How would the confidence interval change if we used a 90% confidence level?
8. How can this method be applied to other measurements in archaeology, such as pottery thickness or weight?

**Tip:** Confidence intervals provide a range of values within which we expect the true population parameter to fall, given a certain level of confidence. The width of this interval depends on the variability of the data, the sample size, and the confidence level chosen.

Learn how to calculate the sample mean, sample standard deviation, and an 80% confidence interval for archaeological rim diameters using given data. Understand the application of descriptive statistics in archaeological research. Suitable for advanced high school students.

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