Question

Teacher raises Refer to Exercise $23$ Suppose each teacher receives a $5\%$ raise instead of a $\$1000$ raise. What effect will this raise have on each of the following characteristics of the

resulting salary distribution?

a. Shape

b. Median

c. Interquartile range (IQR)

Short answer

Part (a) Shape will remain unaffected.

Part (b) Median is increased by $5\%$

Part (c) Interquartile range increases by $5\%$

Step-by-step solution

Part (a) Step 1: Given information

Each teacher receives a $5\%$ raise instead of a $\$1000$ raise.

Every teacher gets a $5\%$ raise.

Part (a) Step 2: Explanation

Instead of $100$percent, the new salary is $105$percent (or $1.05$) of the old salary. As a result, each payment must be multiplied by $1.05$When every value in the data is multiplied by $1.05$, the structure of the distribution remains unchanged since the associations between each data pair remain unchanged. As a result, the new salary distribution shape is identical to the old salary distribution shape.

Part (b) Step 1: Explanation

Instead of $100$ percent, the new salary is $105$ percent (or $1.05$) of the old salary. As a result, each payment must be multiplied by $1.05$ The Median is a measure of the center, as we all know. When every value in the data is multiplied by $1.05$, the distribution's center is multiplied by $1.05$ as well. As a result, the Median is likewise multiplied by $1.05$ We can also argue that the median has risen by $5\%$

Part (c) Step 1: Explanation

As a result, the new salary is $105$percent (or $1.05$) higher than the previous one, which was $100$percent.

As a result, each pay must be multiplied by $1.05$

The interquartile range is a measure of variability, as we all know.

When every value in the data is multiplied by $1.05$, the distribution's variability is multiplied by $1.05$ as well.

As a result, the interquartile range is multiplied by $1.05$

We may also remark that the interquartile range has expanded by $5\%$