Refer to the accompanying data display that results from a simple random sample of times​ (minutes) between eruptions of the Old Faithful geyser. The confidence level of​ 95% was used. Complete parts​ (a) and​ (b) below.
TInterval
​(85.74,91.76)
x overbarequals88.75
Sxequals8.897431411
nequals36
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Part 1
a. Express the confidence interval in the format that uses the​ "less than" symbol. Round the confidence interval limits given that the original times are all rounded to one decimal place.

enter your response here minless thanmuless than

enter your response here min
​(Round to two decimal places as​ needed.)
Part 2
b. Identify the best point estimate of mu and the margin of error.
The point estimate of mu is

enter your response here minutes.
​(Round to two decimal places as​ needed.)
Part 3
The margin of error is Eequals

enter your response here minutes.
​(Round to two decimal places as​ needed.)

Given the data display from a simple random sample of times (minutes) between eruptions of the Old Faithful geyser, let's address the tasks step by step.

### Part 1a: Express the Confidence Interval

The confidence interval given is \((85.74, 91.76)\). To express this in the "less than" format:
\[ 85.74 \, \text{min} < \mu < 91.76 \, \text{min} \]

So, the response in the correct format rounded to two decimal places is:
\[ 85.74 \, \text{min} < \mu < 91.76 \, \text{min} \]

### Part 2b: Identify the Best Point Estimate and Margin of Error

The point estimate of \(\mu\) is given by the sample mean \(\overline{x}\):
\[ \overline{x} = 88.75 \, \text{minutes} \]

The margin of error \(E\) can be calculated using the confidence interval limits. It is half the width of the confidence interval:
\[ E = \frac{91.76 - 85.74}{2} = \frac{6.02}{2} = 3.01 \, \text{minutes} \]

So, the point estimate of \(\mu\) and the margin of error \(E\) are:
\[ \text{Point estimate of } \mu = 88.75 \, \text{minutes} \]
\[ \text{Margin of error } E = 3.01 \, \text{minutes} \]

### Summary
1. Confidence interval: \(85.74 \, \text{min} < \mu < 91.76 \, \text{min}\)
2. Point estimate of \(\mu\): \(88.75 \, \text{minutes}\)
3. Margin of error \(E\): \(3.01 \, \text{minutes}\)

Do you want any details or have any questions?

Here are some related questions to expand your understanding:

1. How do you interpret the confidence interval in practical terms?
2. What is the significance of the margin of error in a confidence interval?
3. How does the sample size affect the confidence interval?
4. What assumptions are made when constructing a confidence interval for a population mean?
5. How would changing the confidence level to 99% affect the interval?
6. What is the formula for calculating the margin of error for a confidence interval?
7. How can you use the standard deviation to understand the spread of eruption times?
8. Why is it important to use a random sample when constructing a confidence interval?

**Tip:** Always ensure your data is rounded consistently to avoid errors in statistical calculations.

Learn how to interpret confidence intervals and calculate point estimates with a detailed example from a simple random sample of Old Faithful geyser eruption times. This problem covers key statistical concepts including confidence intervals, point estimates, and margin of error.

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