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.


The provided confidence interval for the time between eruptions of the Old Faithful geyser is \( (85.74, 91.76) \). To express this confidence interval in the format that uses the "less than" symbol, we can write:

\[
85.7 < \mu < 91.8
\]

### Steps:
1. The original confidence interval is \( (85.74, 91.76) \), but we are rounding to one decimal place because the original times are rounded to one decimal place.
2. After rounding:
   - Lower limit: \( 85.74 \to 85.7 \)
   - Upper limit: \( 91.76 \to 91.8 \)
3. Therefore, the confidence interval can be expressed as:
   \[
   85.7 < \mu < 91.8
   \]

Here, \( \mu \) represents the true mean time between eruptions.

Would you like more details on this, or do you have any other questions?

### Relative Questions:
1. What does the confidence level of 95% signify in this context?
2. How do you interpret the confidence interval for the population mean?
3. What are the possible reasons for using a T-interval instead of a Z-interval?
4. How would the confidence interval change if the sample size increased?
5. What assumptions must be met for the confidence interval to be valid?

### Tip:
When rounding confidence interval limits, always ensure the number of decimal places matches the precision of the original data points.

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) x̄=88.75 Sx=8.897431411 n=36. 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.

Learn how to calculate and interpret the 95% confidence interval (85.7 < µ < 91.8) for the time between eruptions of the Old Faithful geyser. This problem covers key statistical concepts like T-intervals, the central limit theorem, and confidence levels, with a detailed solution for students studying undergraduate or AP Statistics.

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