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.

85.74 minless thanmuless than

91.76 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

88.75 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.)

Let's work through each part of the problem step by step.

### Part 1: Express the Confidence Interval Using the "Less Than" Symbol

Given:
- The confidence interval is \( (85.74 \text{ min}, 91.76 \text{ min}) \).

In the format using the "less than" symbol, the confidence interval is:
\[ 85.74 \text{ min} < \mu < 91.76 \text{ min} \]

Since the original times are already rounded to two decimal places, the confidence interval remains:
\[ 85.74 \text{ min} < \mu < 91.76 \text{ min} \]

### Part 2: Identify the Best Point Estimate of \(\mu\) and the Margin of Error

#### Point Estimate of \(\mu\)
The best point estimate of \(\mu\) is given as \( \bar{x} = 88.75 \text{ minutes} \).

#### Margin of Error \( E \)
The margin of error \( E \) can be calculated using the formula:
\[
E = \text{Upper limit of the confidence interval} - \text{Point estimate}
\]
Substituting the values:
\[
E = 91.76 \text{ min} - 88.75 \text{ min} = 3.01 \text{ min}
\]

### Summary of Answers:

1. **Confidence Interval**:  
   \( 85.74 \text{ min} < \mu < 91.76 \text{ min} \)

2. **Point Estimate of \(\mu\)**:  
   \( \mu = 88.75 \text{ minutes} \)

3. **Margin of Error**:  
   \( E = 3.01 \text{ minutes} \)

Would you like any more details or explanations on this problem? 

Here are some related questions you might find helpful:

1. How is the confidence interval related to the sample mean and standard deviation?
2. What is the significance of a 95% confidence level?
3. How do you interpret the margin of error in a confidence interval?
4. How would the confidence interval change if the sample size increased?
5. What assumptions are necessary for constructing a T-interval?
6. How do you calculate a confidence interval using a Z-interval instead of a T-interval?
7. What would happen to the confidence interval if the confidence level were increased to 99%?
8. How does the variability of the sample (standard deviation) affect the width of the confidence interval?

**Tip:** Always ensure that the sample size is sufficiently large when using the T-distribution for confidence intervals, as it assumes that the sample is representative of the population.

Explore how to calculate a confidence interval for the eruption times of the Old Faithful geyser with a 95% confidence level. Understand the point estimate of the mean and the margin of error in statistical analysis. Suitable for undergraduate students and learners interested in statistics.

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