Unit E: Statistics and Probability

Chapter 1: Statistics

Practice

Instructions: Click the Download File button to download a printable PDF of the questions. Answer each of the following practice questions on a separate piece of paper. Step by step solutions are provided under the Solutions tab. You will learn the material more thoroughly if you complete the questions before checking the answers.

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Practice
Solutions

Shanley is finishing her basement. She has all the required supplies, but she needs to hire workers to install the light fixtures, trim around the doors and windows, and carpeting.

Item to be Installed	Number of Hours Needed	Cost per Hour
light fixtures	2	$50
trim	20	$35
carpeting	9	$40

Which formula should be used to find the average cost per hour: the mean formula or the weighted mean formula?
Calculate the average cost of labour per hour.

Françpois buys $35 items from a store that sells items for a maximum price of $5.00.

The receipt listed the following prices:

$1, $1, $5, $2, $1, $1, $1, $5, $5, $3, $5, $2, $2, $3, $2, $5, $1, $3, $3, $3, $1, $1, $1, $5, $5, $3, $2, $2, $3, $2, $5, $1, $1, $1, $5
1. Which formula should be used to find the average cost per item: the mean formula or the weight mean formula?
2. Calculate the average cost per item.

Assessment	Weight	Carmen's Mark
Assignments	40%	88.1%
Course reviews	10%	82.5%
Exams	50%	75.0%

Carmen would like to calculate her final mark in her statistics course.

Which formula should Carmen use to calculate her mark: the mean formula or the weighted mean formula? Explain.
Use the weighted mean to calculate Carmen's final mark.

In chemistry, the weight of an atom is measured in a tiny unit called the atomic mass unit, which has an abbreviation of amu. The average mass of an atom is calculated using the weighted mean formula.

The symbol for chromium is Cr. Chromium has four possible weights: 50 amu, 52 amu, 53 amu, and 54 amu

Mass of Chromium Atom (amu)	Percent Occurrence (%)
50	4.3
52	83.8
53	9.5
54	2.4

Calculate the average weight of a chromium atom. Answer to the nearest hundredth.

At Prairie Enterprises, 70% of employees are software developers, 20% are engineers, and 10% are managers. Software developers make an average of $50 K per year, and engineers make an average of $80 K per year. If the average salary at the company is $65 K, what is the average annual salary of the managers?

Note: "K" stands for thousand.

Question 1

Shanley is finishing her basement. She has all the required supplies, but she needs to hire workers to install the light fixtures, trim around the doors and windows, and carpeting.

Item to be Installed	Number of Hours Needed	Cost per Hour
light fixtures	2	$50
trim	20	$35
carpeting	9	$40

Which formula should be used to find the average cost per hour: the mean formula or the weighted mean formula?
Calculate the average cost of labour per hour.

Solutions

It is easier to use the weighted mean formula since the data is grouped.
$\begin{array}{rcl} \bar{x} & = & \frac{sum of the values in the data set}{total number of values in the data set} \\ = & \frac{w_{1} (x_{1}) + w_{2} (x_{2}) + w_{3} (x_{3})}{w_{1} + w_{2} + w_{3}} \\ = & \frac{2 (50) + 20 (35) + 9 (40)}{2 + 20 + 9} \\ = & \frac{100 + 700 + 360}{31} \\ = & \frac{1 160}{31} \\ = & 37.42 \end{array}$

Note: The second line of the formula does not need to be included for your work on the assignment, so it will be excluded from the remaining practice questions.

The average cost of labour is $37.42 per hour.

Question 2

Françpois buys $35 items from a store that sells items for a maximum price of $5.00.

The receipt listed the following prices:

$1, $1, $5, $2, $1, $1, $1, $5, $5, $3, $5, $2, $2, $3, $2, $5, $1, $3, $3, $3, $1, $1, $1, $5, $5, $3, $2, $2, $3, $2, $5, $1, $1, $1, $5

Which formula should be used to find the average cost per item: the mean formula or the weight mean formula?
Calculate the average cost per item.

Solutions

The mean formula can be used to find the average cost per item, but it may be easier to use the weighted mean formula since the data set of 35 items has five possible values that are repeated.

Using the weighted mean formula:

Item Cost ($)	Number of Items
1	12
2	7
3	7
5	9

\begin{array}{rcl} \bar{x} & = & \frac{sum of the values in the data set}{total number of values in the data set} \\ = & \frac{12 (1) + 7 (2) + 7 (3) + 9 (5)}{35} \\ = & \frac{12 + 14 + 21 . + 45}{12 + 7 + 9} \\ = & \frac{92}{35} \\ = & 2.63 \end{array}

The average cost is $2.63 per item.

Using the mean formula:

\begin{array}{rcl} \bar{x} & = & \frac{sum of the values in the data set}{total number of values in the data set} \\ = & \frac{1 + 1 + 5 + 2 + 1 + 1 + 1 + 5 + 5 + 3 + 5 + 2 + 2 + 3 + 2 + 5 + 1 + 3 + 3 + 3 + 1 + 1 + 1 + 5 + 5 + 3 + 2 + 2 + 3 + 2 + 5 + 1 + 1 + 1 + 5}{35} \\ = & \frac{92}{35} \\ = & 2.63 \end{array}

The average cost is $2.63 per item.

Question 3

Assessment	Weight	Carmen's Mark
Assignments	40%	88.1%
Course reviews	10%	82.5%
Exams	50%	75.0%

Carmen would like to calculate her final mark in her statistics course.

Which formula should Carmen use to calculate her mark: the mean formula or the weighted mean formula? Explain.
Use the weighted mean to calculate Carmen's final mark.

Solutions

Carmen should use the weighted mean since the values in the set of data are weighted differently.
$\begin{array}{rcl} \bar{x} & = & \frac{sum of the values in the data set}{total number of values in the data set} \\ = & \frac{40 (88.1) + 10 (82.5) + 50 (75.0)}{40 + 10 + 50} \\ = & \frac{3 524 + 825 + 3 750}{100} \\ = & \frac{8 099}{100} \\ = & 81.0 \end{array}$

Carmen's final mark in statistics is 81%.

Question 4

In chemistry, the weight of an atom is measured in a tiny unit called the atomic mass unit, which has an abbreviation of amu. The average mass of an atom is calculated using the weighted mean formula.

The symbol for chromium is Cr. Chromium has four possible weights: 50 amu, 52 amu, 53 amu, and 54 amu

Mass of Chromium Atom (amu)	Percent Occurrence (%)
50	4.3
52	83.8
53	9.5
54	2.4

Calculate the average weight of a chromium atom. Answer to the nearest hundredth.

Solution

\begin{array}{rcl} \bar{x} & = & \frac{sum of the values in the data set}{total number of values in the data set} \\ = & \frac{4.3 (50) + 83.85 (52) + 9.5 (53) + 2.4 (54)}{4.3 + 83.8 + 9.5 + 2.4} \\ = & \frac{215 + 4 357.6 + 503.5 + 129.6}{100} \\ = & \frac{5 205.7}{100} \\ = & 52.06 \end{array}

The average weight of a chromium atom is 52.06 amu.

Question 5

At Prairie Enterprises, 70% of employees are software developers, 20% are engineers, and 10% are managers. Software developers make an average of $50 K per year, and engineers make an average of $80 K per year. If the average salary at the company is $65 K, what is the average annual salary of the managers?

Note: "K" stands for thousand.

Solution

The weighted mean formula should be used since the values in the set of data are weighted differently. Let x = the salary of the managers.

\begin{array}{rcl} \bar{x} & = & \frac{sum of the values in the data set}{total number of values in the data set} \\ 65 & = & \frac{70 (50) + 20 (80) + 10 (x)}{70 + 20 + 10} \\ 65 & = & \frac{3 500 + 1 600 + 10 (x)}{100} \\ 65 & = & \frac{5 100 + 10 (x)}{100} \\ 65 \times 100 & = & (\frac{5 100 + 10 x}{100}) \times 100 \\ 6 500 & = & 5 100 + 10 x \\ 6 500 - 5 100 & = & 5 100 + 10 x - 5 100 \\ 1 400 & = & 10 x \\ \frac{1 400}{10} & = & \frac{10 x}{10} \\ 140 & = & x \end{array}

The managers earn an average salary of $140 K, or $140 000, per year.