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Mr. Smith gave an exam to his students. The table shows their scores over 100 in decimal form. Given the table of 100 scores of students from the exam, how many scores fall within two standard deviations of the mean? The standard deviation is σ=0.13 and the mean is x¯¯¯=0.45 . (1 point) Responses 86 86 58 58 77 77 96 96

1 answer

To determine how many scores fall within two standard deviations of the mean, we first need to calculate the range that captures this criterion.

  1. Calculate the boundaries:

    • The mean \( \bar{x} = 0.45 \)
    • The standard deviation \( \sigma = 0.13 \)

    Now, calculate two standard deviations from the mean:

    • Lower boundary: \( \bar{x} - 2\sigma = 0.45 - 2(0.13) = 0.45 - 0.26 = 0.19 \)
    • Upper boundary: \( \bar{x} + 2\sigma = 0.45 + 2(0.13) = 0.45 + 0.26 = 0.71 \)

    Thus, the range of scores within two standard deviations of the mean is from 0.19 to 0.71.

  2. Count the scores in the given range: You have provided a set of scores, and we will count which scores lie within this range. The provided scores are:

    • 86
    • 86
    • 58
    • 58
    • 77
    • 77
    • 96
    • 96

    Now, let's check each score:

    • 86: Not within (0.19 to 0.71)
    • 86: Not within (0.19 to 0.71)
    • 58: Not within (0.19 to 0.71)
    • 58: Not within (0.19 to 0.71)
    • 77: Not within (0.19 to 0.71)
    • 77: Not within (0.19 to 0.71)
    • 96: Not within (0.19 to 0.71)
    • 96: Not within (0.19 to 0.71)

None of the provided scores fall within the range of 0.19 to 0.71.

Therefore, the number of scores that fall within two standard deviations of the mean is 0.