To calculate the standard deviation $\sigma_X$ of the temperature distribution, we'll use the formula for standard deviation in a discrete probability distribution:

$\sigma_X = \sqrt{\sum (x_i - \mu)^2 \cdot P(x_i)}$

Where:

• $x_i$ are the temperature values
• $P(x_i)$ are the corresponding probabilities
• $\mu$ is the mean (expected value) of the distribution

Step 1: Calculate the Mean $\mu$

The mean $\mu$ for a discrete probability distribution is calculated as:

$\mu = \sum x_i \cdot P(x_i)$

Using the given dataTo calculate the standard deviation $\sigma_X$ of the temperature distribution, we'll use the formula for standard deviation in a discrete probability distribution:

$\sigma_X = \sqrt{\sum (x_i - \mu)^2 \cdot P(x_i)}$

Where:

• $x_i$ are the temperature values
• $P(x_i)$ are the corresponding probabilities
• $\mu$ is the mean (expected value) of the distribution

Step 1: Calculate the Mean $\mu$

The mean $\mu$ for a discrete probability distribution is calculated as:

$\mu = \sum x_i \cdot P(x_i)$

Using the given

$\mu = (540 \cdot 0.1) + (545 \cdot 0.1) + (550 \cdot 0.3) + (555 \cdot 0.25) + (560 \cdot 0.25)$

$\mu = 54 + 54.5 + 165 + 138.75 + 140 = 552.25$

So, the mean $\mu$ is 552.25°C.

Step 2: Calculate the Variance

The variance is:

$\sigma_X^2 = \sum (x_i - \mu)^2 \cdot P(x_i)$

Now we'll calculate each term in this sum:

1. For $x_1 = 540$: $(540 - 552.25)^2 = (-12.25)^2 = 150.0625$ Contribution to variance: $150.0625 \cdot 0.1 = 15.00625$

2. For $x_2 = 545$: $(545 - 552.25)^2 = (-7.25)^2 = 52.5625$ Contribution to variance: $52.5625 \cdot 0.1 = 5.25625$

3. For $x_3 = 550$: $(550 - 552.25)^2 = (-2.25)^2 = 5.0625$ Contribution to variance: $5.0625 \cdot 0.3 = 1.51875$

4. For $x_4 = 555$: $(555 - 552.25)^2 = 2.75^2 = 7.5625$ Contribution to variance: $7.5625 \cdot 0.25 = 1.890625$

5. For $x_5 = 560$: $(560 - 552.25)^2 = 7.75^2 = 60.0625$ Contribution to variance: $60.0625 \cdot 0.25 = 15.015625$

Now, sum these contributions:

$\sigma_X^2 = 15.00625 + 5.25625 + 1.51875 + 1.890625 + 15.015625 = 38.6875$

Step 3: Calculate the Standard Deviation

Finally, take the square root of the variance:

$\sigma_X = \sqrt{38.6875} = 6.220$

So, the standard deviation $\sigma_X$ is 6.220°C (rounded to the nearest thousandth).

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Would you like further clarification on any part of this solution?

Here are some related questions you might find helpful:

1. How is the mean calculated for discrete probability distributions?
2. What is the significance of variance in a data set?
3. Can standard deviation be negative?
4. How does the shape of the distribution relate to standard deviation?
5. What are other ways to measure spread in a data set?

Tip: When interpreting standard deviation, a smaller value indicates that the data points are closer to the mean, while a larger value indicates more spread.