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In descriptive statistics, the mean is a measure often referred to as the "average." It gives us a central value of a dataset and is calculated by summing all values in the dataset and then dividing by the number of values.

To find the mean of the diners' responses, start by adding together all the amounts they reported:

• Sum: \(7.50 + 25.00 + 10.00 + 10.00 + 7.50 + 8.25 + 9.00 + 5.00 + 15.00 + 8.00 + 7.25 + 7.50 + 8.00 + 7.00 + 12.00 = 136.00\)

The next step involves dividing this total sum by the number of responses, which is 15:

• \(\text{Mean} = \frac{136.00}{15} = 9.07\)

Thus, the mean cost diners are willing to pay is approximately \( \$9.07 \). The mean gives us an idea of the "typical" amount diners are considering, assuming all amounts contribute equally to this picture.

The median represents the middle value of a dataset when values are arranged in order. Unlike the mean, the median provides us with the most central point of data, free from the influence of outliers.

To find the median in the diners' data, arrange all amounts in ascending order:

• \([5.00, 7.00, 7.25, 7.50, 7.50, 7.50, 8.00, 8.00, 8.25, 9.00, 10.00, 10.00, 12.00, 15.00, 25.00]\)

Once sorted, find the middle number. Since there are 15 numbers, the median is the 8th number:

• \( \text{Median} = 8.00 \)

This median of \( \$8.00 \) indicates that half of the diners would pay less than or equal to this amount for a meal, reflecting what is most typical without distortion by extreme values.

Analyzing the shape of a distribution provides insight into the data's characteristics and tendencies. By comparing the mean and median, we can understand much about the symmetry or skew of the distribution.

In this exercise, the mean was calculated to be \( \\(9.07 \) while the median is \( \\)8.00 \). Notably, the mean is greater than the median, which often indicates a right (positive) skew in the data distribution.

This pattern occurs when high values, like the particularly large \( \$25.00 \) response, pull the mean upward, more so than the median, which remains unaffected by such outliers.

To determine skewness:

• If the mean > median: expect a right skew.
• If the mean < median: expect a left skew.
• If the mean = median: the distribution is likely symmetric.

Hence, in this scenario, the data distribution leans towards a positive skew, which signifies the presence of higher values affecting the central tendency measure.