Question

a persons metabolic rate is the rate at which the body consumes energy. metabolic rate is important in studies of weight gain, dieting, and exercise. here are the metabolic rates of 7 men who took part in a study of dieting. (the units are calories per 24 hours. these are the same calories used to describe the energy content of foods.) 1792 1666 1362 1614 1460 1867 1439 calculate the standard deviation. (s_x=) calories (round to 2 decimal places.) interpret the standard deviation. in each interpretation, (s_x) represents the value of the standard deviation. on average, the metabolic rate of the seven men who took part in a study of dieting is about (s_x) calories. the range of the middle 50% of the metabolic rates for these seven men is approximately (s_x) calories. the number of calories that each man burns is (s_x) calories away from the mean. the metabolic rate of the seven men who took part in a study of dieting typically varies from the mean by about (s_x) calories.

Explanation:

Step1: Calculate the mean

Let the data set be \(x_1 = 1792\), \(x_2=1666\), \(x_3 = 1362\), \(x_4=1614\), \(x_5 = 1460\), \(x_6=1867\), \(x_7 = 1439\). The number of data points \(n = 7\). The mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}=\frac{1792 + 1666+1362+1614+1460+1867+1439}{7}=\frac{10200}{7}\approx1457.14\).

Step2: Calculate the squared - differences

\((x_1-\bar{x})^2=(1792 - 1457.14)^2=(334.86)^2 = 112121.22\), \((x_2-\bar{x})^2=(1666- 1457.14)^2=(208.86)^2=43622.49\), \((x_3-\bar{x})^2=(1362 - 1457.14)^2=(-95.14)^2 = 9051.62\), \((x_4-\bar{x})^2=(1614-1457.14)^2=(156.86)^2 = 24605.06\), \((x_5-\bar{x})^2=(1460 - 1457.14)^2=(2.86)^2=8.18\), \((x_6-\bar{x})^2=(1867-1457.14)^2=(409.86)^2 = 167985.22\), \((x_7-\bar{x})^2=(1439-1457.14)^2=(-18.14)^2 = 329.06\).

Step3: Calculate the variance

The variance \(s^2=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}=\frac{112121.22+43622.49+9051.62+24605.06+8.18+167985.22+329.06}{6}=\frac{357722.85}{6}\approx59620.48\).

Step4: Calculate the standard deviation

The standard deviation \(s=\sqrt{s^2}=\sqrt{59620.48}\approx244.17\).

Answer:

\(s_x = 244.17\)
The metabolic rate of the seven men who took part in a study of dieting typically varies from the mean by about \(s_x\) calories.