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.