A set of test grades has a mean of 70 and a standard deviation of 6. If the teacher shifts the scores by adding 5 points to each grade, the new mean and standard deviation will be:
A. mean = 75, standard deviation = 11
B. mean = 70, standard deviation = 11
C. mean = 75, standard deviation = 6
Answer (2)
The original set of test grades has a mean of 70 and a standard deviation of 6.
Adding 5 points to each grade shifts the mean by 5, resulting in a new mean of 75.
Adding a constant to each grade does not change the standard deviation, so the new standard deviation remains 6.
The new mean and standard deviation are mean = 75 , standard deviation = 6 .
Explanation
Understanding the Problem We are given a set of test grades with a mean of 70 and a standard deviation of 6. The teacher adds 5 points to each grade, and we need to find the new mean and standard deviation.
Defining the Random Variable Let X be the random variable representing the original test grades. The mean of X is E [ X ] = 70 , and the standard deviation of X is S D [ X ] = 6 .
Defining the New Grades The new grades are represented by the random variable Y = X + 5 . We want to find the mean and standard deviation of Y .
Calculating the New Mean The new mean is E [ Y ] = E [ X + 5 ] = E [ X ] + 5 . Since the original mean E [ X ] = 70 , the new mean is 70 + 5 = 75 .
Calculating the New Standard Deviation The new standard deviation is S D [ Y ] = S D [ X + 5 ] = S D [ X ] . Adding a constant to a random variable does not change its standard deviation.
Final Answer Since the original standard deviation S D [ X ] = 6 , the new standard deviation is also 6. Therefore, the new mean is 75 and the new standard deviation is 6.
Examples
Imagine you are tracking the daily temperature in Celsius, and you decide to convert all the readings to Fahrenheit. Adding a constant to each reading (or multiplying by a constant) is similar to shifting the test scores. Understanding how these shifts affect the mean and standard deviation helps you analyze the data in the new scale without recalculating everything from scratch. This is useful in various fields, such as finance, engineering, and statistics, where data transformations are common.
The new mean of the test grades after adding 5 points to each grade is 75, while the standard deviation remains 6. Hence, the correct answer is option C: mean = 75, standard deviation = 6.
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