Question

the data sets give the number of platinum albums for the five male artists and the five female artists in a country with the most platinum albums through a recent year (platinum albums sell one million units or more ) complete parts (a) through (d) below
c. without calculating, which data set has the greater standard deviation? explain your answer
a. male artists have the greater standard deviation because there is a greater spread in the data for men
b. female artists have the greater standard deviation because there is a lower spread in the data for women
c. male artists have the greater standard deviation because there is a lower spread in the data for men
d. female artists have the greater standard deviation because there is a greater spread in the data for women
d. verify your conjecture from part (c) by calculating the standard deviation for each data set
standard deviation (male artists) =
standard deviation (female artists) =
(round to two decimal places as needed )

Explanation:

Step1: Calculate the mean for male artists

The formula for the mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\). For male artists \(x_1 = 146\), \(x_2=94\), \(x_3 = 81\), \(x_4=74\), \(x_5 = 69\), \(n = 5\).
\(\bar{x}_{male}=\frac{146 + 94+81 + 74+69}{5}=\frac{464}{5}=92.8\)

Step2: Calculate the squared - differences for male artists

\((x_1-\bar{x}_{male})^2=(146 - 92.8)^2=(53.2)^2 = 2830.24\)
\((x_2-\bar{x}_{male})^2=(94 - 92.8)^2=(1.2)^2=1.44\)
\((x_3-\bar{x}_{male})^2=(81 - 92.8)^2=(- 11.8)^2 = 139.24\)
\((x_4-\bar{x}_{male})^2=(74 - 92.8)^2=(-18.8)^2=353.44\)
\((x_5-\bar{x}_{male})^2=(69 - 92.8)^2=(-23.8)^2 = 566.44\)

Step3: Calculate the variance for male artists

The formula for the variance \(s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}\).
\(s_{male}^{2}=\frac{2830.24+1.44 + 139.24+353.44+566.44}{4}=\frac{3890.8}{4}=972.7\)

Step4: Calculate the standard deviation for male artists

\(s_{male}=\sqrt{972.7}\approx31.17\)

Step5: Calculate the mean for female artists

For female artists \(x_1 = 65\), \(x_2=63\), \(x_3 = 57\), \(x_4=52\), \(x_5 = 44\), \(n = 5\)
\(\bar{x}_{female}=\frac{65 + 63+57 + 52+44}{5}=\frac{281}{5}=56.2\)

Step6: Calculate the squared - differences for female artists

\((x_1-\bar{x}_{female})^2=(65 - 56.2)^2=(8.8)^2 = 77.44\)
\((x_2-\bar{x}_{female})^2=(63 - 56.2)^2=(6.8)^2=46.24\)
\((x_3-\bar{x}_{female})^2=(57 - 56.2)^2=(0.8)^2 = 0.64\)
\((x_4-\bar{x}_{female})^2=(52 - 56.2)^2=(-4.2)^2=17.64\)
\((x_5-\bar{x}_{female})^2=(44 - 56.2)^2=(-12.2)^2 = 148.84\)

Step7: Calculate the variance for female artists

\(s_{female}^{2}=\frac{77.44+46.24 + 0.64+17.64+148.84}{4}=\frac{290.8}{4}=72.7\)

Step8: Calculate the standard deviation for female artists

\(s_{female}=\sqrt{72.7}\approx8.53\)

Answer:

Standard deviation (Male artists) \(\approx31.17\)
Standard deviation (Female artists) \(\approx8.53\)