Dean Chalmers and Julian Gilbey Solutions for Chapter: Measures of Variation, Exercise 12: END-OF-CHAPTER REVIEW EXERCISE 3

A shop has in its stock$80$ rectangular celebrity posters. All of these posters have a width to height ratio of $1:\sqrt{2},$ and their mean perimeter is $231.8 \mathrm{cm}.$ Given that the sum of the squares of the widths is $200120{ \mathrm{cm}}^2,$ find the standard deviation of the widths of the posters.

At a village fair, visitors were asked to guess how many sweets are in a glass jar. The best six guesses were $180, 211, 230, 199, 214$ and $166.$

Show that the mean of these guesses is $200,$ and use $\mathrm{SD}=\sqrt{\frac{\Sigma (x-\overline{x}{)}^2}{n}}$ to calculate the standard deviation.

At a village fair, visitors were asked to guess how many sweets are in a glass jar. The best six guesses were $180, 211, 230, 199, 214$ and $166.$

The jar actually contained$202$sweets. Without further calculation, write down the mean and the standard deviation of the errors made by these six visitors. Explain why no further calculations are required to do this.

The number of women in senior management positions at a number of companies was investigated. The number of women at each of the $25$ service companies and at each of the $16$ industrial companies are denoted by $w_{\mathrm{s}}$ and $w_1,$ respectively. The findings are summarised by the totals:
$\Sigma {\left(w_{\mathrm{s}}-5\right)}^2=28,\Sigma \left(w_{\mathrm{s}}-5\right)=15,\Sigma {\left(w_1-3\right)}^2=12$ and $\Sigma \left(w_1-3\right)=-4$

Show that there are, on average, more than twice as many women in senior management positions at the service companies than at the industrial companies.

The number of women in senior management positions at a number of companies was investigated. The number of women at each of the $25$ service companies and at each of the $16$ industrial companies are denoted by $w_s$ and $w_1,$ respectively. The findings are summarised by the totals:
$\Sigma {\left(w_{\mathrm{s}}-5\right)}^2=28,\Sigma \left(w_{\mathrm{s}}-5\right)=15,\Sigma {\left(w_1-3\right)}^2=12$ and $\Sigma \left(w_1-3\right)=-4$

Show that $\Sigma w_{\mathrm{s}}^2\ne {\left(\Sigma w_{\mathrm{s}}\right)}^2$ and that $\Sigma w_1^2\ne {\left(\Sigma w_{\mathrm{I}}\right)}^2.$

The number of women in senior management positions at a number of companies was investigated. The number of women at each of the $25$ service companies and at each of the $16$ industrial companies are denoted by $w_{\mathrm{s}}$ and $w_1,$ respectively. The findings are summarised by the totals:
$\Sigma {\left(w_{\mathrm{s}}-5\right)}^2=28,\Sigma \left(w_{\mathrm{s}}-5\right)=15,\Sigma {\left(w_1-3\right)}^2=12$ and $\Sigma \left(w_1-3\right)=-4$

Find the standard deviation of the number of women in senior management positions at all of these service and industrial companies together.

The ages, $a$ years, of the five members of the boy-band AlphaArise are such that $\Sigma {\left(a-21\right)}^2=11.46$ and $\Sigma \left(a-21\right)=-6.$
The ages, $b$ years, of the seven members of the boy-band BetaBeat are such that $\Sigma {\left(b-18\right)}^2=10.12$ and $\Sigma \left(b-18\right)=0.$

Show that the difference between the mean ages of the boys in the two bands is $1.8$ years.

The ages, $a$ years, of the five members of the boy-band AlphaArise are such that $\Sigma {\left(a-21\right)}^2=11.46$ and $\Sigma \left(a-21\right)=-6.$
The ages, $b$ years, of the seven members of the boy-band BetaBeat are such that $\Sigma {\left(b-18\right)}^2=10.12$ and $\Sigma \left(b-18\right)=0.$

Find the variance of the ages of the $12$ members of these two bands.