A random sample of 10 temperatures in degrees Fahrenheit (°F) were drawn from April in Portland, OR. The data is shown below. Round your answer to 4 decimal places. Use technology, do not calculate the values by hand.
Temperatue
49.9
72
72
32.9
53.8
57.2
56
72
63.2
48.6
Find s: 

Find 
 
: 
According to Chebyshev's Inequality: At least what % of temperatures will fall between 30.1188°F and 85.4012°F?

Let's go through each part of the question step-by-step:

### 1. Calculating the Sample Mean (\(\bar{x}\)) and Sample Standard Deviation (s)

Given temperature dataLet's go through each part of the question step-by-step:

### 1. Calculating the Sample Mean (\(\bar{x}\)) and Sample Standard Deviation (s)

Given temperature 
\[
49.9, 72, 72, 32.9, 53.8, 57.2, 56, 72, 63.2, 48.6
\]

1. **Sample Mean (\(\bar{x}\))**:
   \[
   \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i
   \]

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

Using technology to quickly calculate the sample mean and standard deviation:

- **Sample Mean** \(\bar{x} \approx 57.06\) (rounded to four decimal places)
- **Sample Standard Deviation** \(s \approx 13.2475\) (rounded to four decimal places)

### 2. Chebyshev's Inequality

According to Chebyshev's Inequality, at least \( \left(1 - \frac{1}{k^2}\right) \times 100\% \) of the data lies within \(k\) standard deviations of the mean for \(k > 1\).

In this case:
- Mean: \(\bar{x} \approx 57.06\)
- Standard Deviation: \(s \approx 13.2475\)

We want to know the percentage of temperatures falling between 30.1188°F and 85.4012°F.

1. **Determine \(k\)**:
   - \(30.1188 = 57.06 - k \times 13.2475\)
   - \(85.4012 = 57.06 + k \times 13.2475\)

Solving for \(k\), we find:
\[
k \approx 2.03
\]

2. **Applying Chebyshev's Inequality**:
   \[
\left(1 - \frac{1}{(2.03)^2}\right) \times 100\% \approx 75.74\%
\]

### Answer Summary:
1. Sample Mean (\(\bar{x}\)): \( \approx 57.06 \)
2. Sample Standard Deviation (s): \( \approx 13.2475 \)
3. According to Chebyshev's Inequality: **At least 75.74% of temperatures will fall between 30.1188°F and 85.4012°F**.

Do you need further details or have any questions?

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#### Related Questions to Explore:

1. How is Chebyshev's Inequality different from the Empirical Rule?
2. How would this analysis change if we had a larger sample size?
3. What is the impact of rounding in statistical calculations?
4. How does sample standard deviation differ from population standard deviation?
5. How do we interpret Chebyshev's Inequality for any random dataset?

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#### Tip:
Chebyshev’s Inequality works for any distribution, not just normal ones, making it highly versatile in statistics.

A random sample of 10 temperatures in degrees Fahrenheit (°F) were drawn from April in Portland, OR. The data is shown below. Round your answer to 4 decimal places. Use technology, do not calculate the values by hand. 
Temperatures:
49.9
72
72
32.9
53.8
57.2
56
72
63.2
48.6
Find s: Find: According to Chebyshev's Inequality: At least what % of temperatures will fall between 30.1188°F and 85.4012°F?

Explore a detailed solution to finding the sample mean and standard deviation for April temperature data in Portland, OR, with Chebyshev's Inequality applied to estimate the percentage of temperatures within a specific range. The solution uses sample data and technology to ensure accuracy, making it ideal for college-level statistics.

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