Chapter 1: Q 102. (page 76)

Healthy fast food?Here is a dotplot of the amount of fat (to the nearest gram) in 12 different hamburgers served at a fast-food restaurant:
(a) The distribution of fat content has a mean of $\bar{x} = 22.83$ grams and a standard deviation of $s_{x} = 9.06$ grams. Interpret the standard deviation.
(b) Suppose the restaurant replaces the burger that has 22 grams of fat with a new burger that has 35 grams of fat. How would this affect the mean and the standard deviation? Justify your answer.

Short Answer

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Part (a) The fat content of the 12 different hamburgers in the sample is expected to vary by 9.06 grams on average, compared to the mean of 22.83 grams.

Part (b) Increase in the mean and the standard deviation rise.

Step by step solution

Part (a) Step 1. Given information.

The given mean and standard deviation are 22.83 and 9.06respectively.

Part (a) Step 2. Interpret the standard deviation.

The standard deviation is a measure of how much the data values deviate from the mean.

The fat content contained in 12 different hamburgers served at a fast-food restaurant in the sample is expected to vary by 9.06 grams on average, compared to the mean of 22.83 grams.

Part (b) Step 1. Affect on mean and the standard deviation.

The data value 22 is replaced with the data value 35.

Because the mean is the total of all data values divided by the number of values when 22 is replaced by 35, the sum of all data values increases, and the mean increases as well.

The standard deviation is a measure of how much the data values deviate from the mean. Because 35 is substantially greater than the original mean of 22.83, we expect the new mean to be much larger as well.

As a result, the standard deviation rises, as 35 moves further away from the mean.