Questions: A survey of 10 fast-food restaurants noted the number of calories in a mid-sized hamburger. The results are given in the table below. Calories in a Mid-Sized Hamburger 514, 508, 502, 497, 495, 507, 457, 478, 464, 515 Find the mean and sample standard deviation of these data. Round to the nearest hundredth.

A survey of 10 fast-food restaurants noted the number of calories in a mid-sized hamburger. The results are given in the table below. Calories in a Mid-Sized Hamburger 514, 508, 502, 497, 495, 507, 457, 478, 464, 515 Find the mean and sample standard deviation of these data. Round to the nearest hundredth.
Transcript text: A survey of 10 fast-food restaurants noted the number of calories in a mid-sized hamburger. The results are given in the table below. Calories in a Mid-Sized Hamburger 514, 508, 502, 497, 495, 507, 457, 478, 464, 515 Find the mean and sample standard deviation of these data. Round to the nearest hundredth.
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Solution

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Solution Steps

Step 1: Calculate the Mean

To find the mean \( \mu \) of the calorie counts in mid-sized hamburgers, we use the formula:

\[ \mu = \frac{\sum_{i=1}^N x_i}{N} \]

where \( N \) is the number of observations and \( x_i \) are the individual calorie counts. The sum of the calorie counts is:

\[ \sum_{i=1}^{10} x_i = 514 + 508 + 502 + 497 + 495 + 507 + 457 + 478 + 464 + 515 = 4937 \]

Thus, the mean is calculated as:

\[ \mu = \frac{4937}{10} = 493.7 \]

Step 2: Calculate the Sample Standard Deviation

The sample standard deviation \( s \) is calculated using the formula:

\[ s = \sqrt{\frac{\sum (x_i - \mu)^2}{n-1}} \]

First, we find the variance \( \sigma^2 \):

\[ \sigma^2 = \frac{\sum (x_i - \mu)^2}{n-1} = 422.68 \]

Then, the sample standard deviation is:

\[ s = \sqrt{422.68} \approx 20.56 \]

Final Answer

The mean and sample standard deviation of the calorie counts are:

\[ \text{Mean } \mu = 493.7 \] \[ \text{Sample Standard Deviation } s \approx 20.56 \]

Thus, the final answers are:

\(\boxed{\mu = 493.7}\)

\(\boxed{s \approx 20.56}\)

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