Averages and Probability

Examples

Solution

Example 5.49 - Solution

Write the formula for the mean:	$\text{mean} = \dfrac{ \text{sum of all the numbers} } {n}$
Write the sum of the numbers in the numerator.	$\text {mean}=\dfrac{8+12+15+9+6}{n}$
Count how many numbers are in the set. There are 5 numbers in the set, so $n=5$ .	$\text {mean}=\dfrac{8+12+15+9+6}{n}$
Add the numbers in the numerator.	$\text {mean}=\dfrac{50}{5}$
Then divide.	$\text {mean}=10$
Check to see that the mean is 'typical': 10 is neither less than 6 nor greater than 15.	The mean is 10.

Example 5.50- Solution

Write the formula for the mean:	$\text{mean} = \dfrac{\text{sum of all the numbers}}{n}$
Write the sum of the numbers in the numerator.	$\text {mean}=\dfrac{16+26+53+56+65+70+93+97}{n}$
Count how many numbers are in the set. Call this $n$ and write it in the denominator.	$\text {mean}=\dfrac{16+26+53+56+65+70+93+97}{8}$
Simplify the fraction.	$\text {mean}=\dfrac{476}{8}$
	$\text {mean}=59.5$

Is 59.5 'typical'? Yes, it is neither less than 16 nor greater than 97. The mean age is 59.5 years.

Example 5.51- Solution

Write the formula for the mean:	$\text{mean} = \dfrac{\text{sum of all the numbers}}{n}$
Count how many numbers are in the set. Call this $n$ and write it in the denominator.	$\text{mean} = \dfrac{\text{sum of all the numbers}}{4}$
Write the sum of the numbers in the numerator.	$\text {mean}=\dfrac{42.75+50.12+41.54+48.15}{4}$
Simplify the fraction.	$\text {mean}=\dfrac{182.56}{4}$
	$\text {mean}=59.5$

Does $45.64 seem 'typical' of this set of numbers? Yes, it is neither less than $41.54 nor greater than $50.12.

The mean cost of her cell phone bill was $45.64

Example 5.52- Solution

List the numbers in order from smallest to largest.	$9, 11, 12, 13, 15, 18, 19$
Count how many numbers are in the set. Call this $n$ .	$n=7$
Is $n$ odd or even?	odd
The median is the middle value.
The middle is the number in the 4th position.	So the median of the data is 13.

List the numbers in order from smallest to largest. $9, 11, 12, 13, 15, 18, 19$

Count how many numbers are in the set. Call this $n$ .
$n=7$

Is $n$ odd or even? odd

The median is the middle value.

The middle is the number in the 4th position.
So the median of the data is 13.

Example 5.53- Solution

Find the median of $83, 79, 85, 86, 92, 100, 76, 90, 88, and 64$ .

List the numbers in order from smallest to largest.
$64, 76, 79, 83, 85, 86, 88, 90, 92, 100$

Count how many numbers are in the set. Call this $n$ .
$n=10$

Is $n$ odd or even?
even

The median is the mean of the two middle values, the 5th and 6th numbers.

Find the mean of $85$ and $86$ . $\text {mean}=\dfrac{85+86}{2}$

$mean=85.5$

Kristen's median score is 85.5.

Find the median of $83, 79, 85, 86, 92, 100, 76, 90, 88, and 64$ .
List the numbers in order from smallest to largest.	$64, 76, 79, 83, 85, 86, 88, 90, 92, 100$
Count how many numbers are in the set. Call this $n$ .	$n=10$
Is $n$ odd or even?	even
The median is the mean of the two middle values, the 5th and 6th numbers.
Find the mean of $85$ and $86$ .	$\text {mean}=\dfrac{85+86}{2}$
	$mean=85.5$
	Kristen's median score is 85.5.

Example 5.54- Solution

The ages are already listed in order. We will make a table of frequencies to help identify the age with the highest frequency.

Now look for the highest frequency. The highest frequency is 7, which corresponds to the age 20. So the mode of the ages in this class is 20 years.

Example 5.55- Solution

List each number with its frequency.

Now look for the highest frequency. The highest frequency is 6, which corresponds to the height 67 inches. So the mode of this set of heights is 67 inches.