You must estimate the mean temperature (in degrees Fahrenheit) with the following sample temperatures:
16.8
17
20
58.2
50.5
28.4
34.4
48.1
19.4
39.4
25
13.2


Find the 90% confidence interval. Enter your answer as an open-interval (i.e., parentheses) accurate to two decimal places.

90% C.I. = 


To calculate the 90% confidence interval for the mean temperature, we can follow these steps:

1. **Find the Sample Mean (\(\bar{x}\)):**
   \[
   \bar{x} = \frac{\sum x}{n}
   \]

2. **Calculate the Sample Standard Deviation (\(s\)):**
   \[
   s = \sqrt{\frac{\sum (x - \bar{x})^2}{n - 1}}
   \]

3. **Determine the Margin of Error (ME):**
   For a 90% confidence interval with \(n = 12\) (small sample), we use the \(t\)-distribution with \(n - 1 = 11\) degrees of freedom. The \(t\)-value for a 90% confidence level is approximately \(t_{0.05, 11} = 1.796\).

   \[
   \text{ME} = t \times \frac{s}{\sqrt{n}}
   \]

4. **Construct the Confidence Interval:**
   The confidence interval is given by:
   \[
   \left( \bar{x} - \text{ME}, \bar{x} + \text{ME} \right)
   \]

Let's calculate each of these values.The sample mean temperature is approximately \( 30.87^\circ \)F, with a sample standard deviation of \( 15.10^\circ \)F.

The 90% confidence interval for the mean temperature is:
\[
(23.04, 38.69)
\]

### 90% C.I. = \( (23.04, 38.69) \)

Would you like further details on the calculation? 

Here are five related questions that may help deepen your understanding:
1. How would the confidence interval change with a 95% confidence level?
2. What impact would a larger sample size have on this confidence interval?
3. How do you interpret the margin of error in a confidence interval?
4. Can you use this confidence interval to make predictions about future temperatures?
5. Why do we use the \( t \)-distribution for small sample sizes instead of the \( z \)-distribution?

**Tip:** Increasing the confidence level (e.g., from 90% to 95%) will widen the confidence interval.

You must estimate the mean temperature (in degrees Fahrenheit) with the following sample temperatures: 16.8, 17, 20, 58.2, 50.5, 28.4, 34.4, 48.1, 19.4, 39.4, 25, 13.2. Find the 90% confidence interval. Enter your answer as an open-interval (i.e., parentheses) accurate to two decimal places.

Learn how to calculate the 90% confidence interval for a mean temperature based on sample data [16.8, 17, 20, 58.2, 50.5, 28.4, 34.4, 48.1, 19.4, 39.4, 25, 13.2]. This guide covers statistical concepts such as the sample mean, sample standard deviation, and margin of error using a t-distribution, suitable for high school and college students.

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