Question

A school system employs teachers at salaries between $38,000 and $70,000. The teachers’ union and school board are negotiating the form of next years’ increase in the salary schedule. Suppose that every teacher is given a $1000 raise.

Which of the following choices correctly describes the effect this raise will have on each of the following characteristics of the resulting distribution of salary:

Shape
Mean and median
Standard deviation and interquartile range (IQR)

Answer 1

Answer:By adding a constant value to every salary amount, the measures of

central tendency are increased by the amount, while the measures of

dispersion, remains the same

The correct responses are;

(a) The shape of the data remains the same

(b) The mean and median are increased by $1,000

(c) The standard deviation and interquartile range remain the same

Reasons:

The given parameters are;

Present teachers salary = Between $38,000 and $70,000

Amount of raise given to every teacher = $1,000

Required:

Effect of the raise on the following characteristics of the data

(a) Effect on the shape of distribution

The outline shape of the distribution will the same but higher by $1,000

(b) The mean of the data is given as follows;

Therefore, following an increase of $1,000, we have;

Therefore, the new mean, is equal to the initial mean increased by 1,000

Median;

Given that all salaries, , are increased by $1,000, the median salary, , is also increased by $1,000

Therefore;

The correct response is that the median is increased by $1,000

(c) The standard deviation, σ, is given by ;

Where;

n = The number of teaches;

Given that, we have both a salary, , and the mean, , increased by $1,000, we can write;

Therefore;

; The standard deviation stays the same

Interquartile range;

The interquartile range, IQR = Q₃ - Q₁

New interquartile range, IQR = (Q₃ + 1000) - (Q₁ + 1000) = Q₃ - Q₁ = IQR

Therefore;

The interquartile range stays the same

Explanation: