Question

the final exam grade distribution for all students in the introductory statistics class at a local community college is displayed in the table, with ( a = 4 ), ( b = 3 ), ( c = 2 ), ( d = 1 ), and ( f = 0 ). let ( x ) represent the grade for a randomly selected student from the class. which statement correctly interprets the standard deviation? the grade for a randomly selected student would typically vary from the expected grade by 1.16. the grade for a randomly selected student would typically vary from the expected grade by 1.08. the mean grade for a randomly selected student would typically vary by 1.16 from the expected grade. the mean grade for a randomly selected student would typically vary by 1.08 from the expected grade

Explanation:

Step1: Recall the definition of standard deviation

The standard deviation of a random variable \(X\) measures the typical deviation of the values of \(X\) from the expected value \(E(X)\).

Step2: Analyze each option

• Option 1: The grade for a randomly - selected student would typically vary from the expected grade by \(1.16\). This is a correct interpretation of the standard deviation. The standard deviation is a measure of the spread of the data around the mean (expected value).
• Option 2: The value \(1.08\) is incorrect. We need to calculate the standard deviation to confirm.
• Option 3: The standard deviation is about the deviation of a single data point (a single student's grade) from the mean, not the mean grade varying.
• Option 4: The value \(1.08\) is incorrect and the description about the mean grade varying is wrong.

Answer:

The grade for a randomly selected student would typically vary from the expected grade by \(1.16\) (the first option).