Question

Construct arguments: The manager of a fast-food restaurant is analyzing the hourly wages of the employees. To make the wages more even, she wants to reduce the spread of the data that represents the wages by adding $1 to each employee’s hourly wage. Is this an effective method to reduce the spread? Justify your response.

Options:
a. Yes; Adding $1 to each employee’s hourly wage has a positive effect on the standard deviation of the data.
b. No; Adding $1 to each employee’s hourly wage has a negative effect on the standard deviation of the data.
c. No; Adding $1 to each employee’s hourly wage has no effect on the standard deviation of the data.
d. Yes; Adding $1 to each employee’s hourly wage has a negative effect on the standard deviation of the data.

Answer 1

Adding $1 to each employee's hourly wage shifts the wage distribution horizontally but doesn't change the distance between the wages; hence, it doesn't alter the spread or standard deviation.

Step-by-step explanation:

The question at hand involves understanding the effects of adding a constant amount to a set of data on its spread or standard deviation. When a constant is added to every value in a data set, this translates into a horizontal shift on the number line for each data point. However, the distance between any two points remains the same. Since standard deviation is a measure of the spread of values around the mean, and since the distances between the data points remain unchanged, the standard deviation does not change. Therefore, the correct answer is 'c. No; Adding $1 to each employee’s hourly wage has no effect on the standard deviation of the data.'

Examples in Real-life Scenarios

Consider salaries of teachers receiving a $3,000 raise. The distribution of the salaries, denoted as X, would shift right by $3,000, which means every teacher's salary would increase by this amount, but the spread measured by standard deviation would not change. Furthermore, if the firm raises the wage to $24 an hour and decides to invest more in machinery, this could result in higher productivity for workers but might not necessarily impact the spread of wages, unless specific changes in pay structure occur as well.