Scaling Data Sample annual salaries (in thousands of dollars) for employees at a company are listed.
42 36 48 51 39 39 42
36 48 33 39 42 45 50
b. Each employee in the sample receives a 5% raise. Find the sample mean and the sample standard deviation for the revised data set.

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Step 1: Calculate the sample mean of the original data set. The formula for the sample mean is \( \bar{x} = \frac{\sum x_i}{n} \), where \( x_i \) represents each data point and \( n \) is the total number of data points. Add all the salaries together and divide by the total number of employees.

Step 2: Calculate the sample standard deviation of the original data set. Use the formula \( s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} \), where \( \bar{x} \) is the sample mean. Subtract the mean from each data point, square the result, sum all squared differences, divide by \( n-1 \), and take the square root.

Step 3: Apply the 5% raise to each salary in the data set. Multiply each salary by 1.05 (since a 5% increase is equivalent to multiplying by 1.05) to create the revised data set.

Step 4: Calculate the sample mean of the revised data set. Since multiplying each data point by a constant scales the mean by the same factor, multiply the original sample mean by 1.05 to find the new mean.

Step 5: Calculate the sample standard deviation of the revised data set. When each data point is multiplied by a constant, the standard deviation is also scaled by the same factor. Multiply the original standard deviation by 1.05 to find the new standard deviation.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Sample Mean

The sample mean is the average of a set of values, calculated by summing all the values and dividing by the number of observations. In this context, it represents the average salary of employees before and after the 5% raise. Understanding how to compute the sample mean is essential for analyzing the overall salary distribution.

Sample Standard Deviation

The sample standard deviation measures the amount of variation or dispersion in a set of values. It indicates how much individual salaries deviate from the sample mean. Calculating the standard deviation is crucial for understanding the spread of salaries and how consistent or varied the salaries are among employees.

Percentage Increase

A percentage increase quantifies how much a value has grown relative to its original amount. In this case, each employee's salary is increased by 5%, which requires adjusting the original salaries before recalculating the mean and standard deviation. Understanding percentage increases is vital for accurately interpreting changes in data sets.

Scaling Data Sample annual salaries (in thousands of dollars) for employees at a company are listed.42 36 48 51 39 39 4236 48 33 39 42 45 50b. Each employee in the sample receives a 5% raise. Find the sample mean and the sample standard deviation for the revised data set.