a The following table represents the students' hourly wages that an amusement park hires for the summer.

Student's Hourly Wages
$$ 16.50$	$$ 8.25$
$$ 8.75$	$$ 8.45$
$$ 8.65$	$$ 8.25$
$$ 9.10$	$$ 9.25$

The given data has a mean of $$ 9.65,$ a median of $$ 8.7,$ and a mode of $$ 8.25 .$ We are asked how these measures will be affected if the park hires another student at an hourly wage of $$ 8.45 .$ Let's analyze each measure of center one at a time.

Mean

The mean of a numerical data set is the sum of the data divided by the number of data values. After hiring the student, we will have

9

data values. Therefore we will divide the sum of the data by

9 .

\overset{x}{ˉ} = \frac{Sum of Values}{Number of Values}

SubstituteValues

Substitute values

\overset{x}{ˉ} = \frac{1 6 . 5 0 + 8 . 7 5 + 8 . 6 5 + 9 . 1 0 + 8 . 2 5 + 8 . 4 5 + 8 . 2 5 + 9 . 2 5 + 8 . 4 5}{9}

▼

Evaluate right-hand side

AddTerms

Add terms

\overset{x}{ˉ} = \frac{8 5 . 6 5}{9}

UseCalc

Use a calculator

\overset{x}{ˉ} = 9.516666 \dots

RoundDec

Round to $2$ decimal place(s)

\overset{x}{ˉ} \approx 9.52

The mean would be about

$ 9.52 .

Comparing the original mean,

$ 9.65,

to the new mean, we can see that the original mean is greater. Let's find the difference of these means.

$ 9.65 - $ 9.52 = $ 0.13

Therefore, after the student is hired the mean decreases by about

$ 0.13 .

Median

The median of a numerical data set is the middle number when the values are written in numerical order. If the data set has an even number of values, the mean of the two middle values will be the median. Similarly, we first need to find the median after the student is hired. Let's write the data in numerical order.

8.25, 8.25, 8.45, 8.45, 8.65, 8.75, 9.10, 9.25, 16.50

We can see that the median is now

$ 8.65 .

Comparing it with the original median,

$ 8.70,

we can see that the original median is greater. Let's find the difference of these medians.

$ 8.70 - $ 8.65 = $ 0.05

Therefore, after the student is hired the median decreases by

$ 0.05 .

Mode

The mode of a data set is the value or values that occur most often. Again, we first need to find the mode for the data with the new wage added. Let's take a look at the given values.

8.25, 8.25, 8.45, 8.45, 8.65, 8.75, 9.10, 9.25, 16.50

This time we have two values that occur most often. Therefore, instead of only having the mode

$ 8.25,

it will now be

$ 8.25

and

$ 8.45 .

Exercise