Math Problem Statement

Sample annual salaries​ (in thousands of​ dollars) for employees at a company are listed. 50  43  46  59  34  34  50  43  46  31  59  50  49
​(a) Find the sample mean and sample standard deviation. ​(b) Each employee in the sample is given a 6​% raise. Find the sample mean and sample standard deviation for the revised data set. ​(c) To calculate the monthly​ salary, divide each original salary by 12. Find the sample mean and sample standard deviation for the revised data set. ​(d) What can you conclude from the results of​ (a), (b), and​ (c)?

Solution

Let's start by solving the questions step-by-step.

(a) Find the Sample Mean and Sample Standard Deviation

Given salaries (in thousands of dollars):

50,43,46,59,34,34,50,43,46,31,59,50,4950, 43, 46, 59, 34, 34, 50, 43, 46, 31, 59, 50, 49

  1. Sample Mean (xˉ\bar{x}):

xˉ=∑xin\bar{x} = \frac{\sum x_i}{n}

where nn is the number of data points.

  1. Sample Standard Deviation (ss):

[ s = \sqrt{\frac{\sum (x_i - \bar{x})^

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Math Problem Analysis

Mathematical Concepts

Descriptive Statistics
Mean
Standard Deviation
Percentage Increase
Scaling Data

Formulas

Sample Mean: x̄ = (Σxᵢ) / n
Sample Standard Deviation: s = √[Σ(xᵢ - x̄)² / (n - 1)]
Revised Salary with 6% Raise: New Salary = Old Salary × 1.06
Monthly Salary: Monthly Salary = Annual Salary / 12

Theorems

Properties of Mean and Standard Deviation under Scaling

Suitable Grade Level

Grades 10-12

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