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18 November, 22:58

The heights of students at a college are normally distributed with a mean of 175 cm and a standard deviation of 6 cm. One might expect in a sample of 1000 students that the number of students with heights less than 163 cm is:

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Answers (2)
  1. 18 November, 23:03
    0
    25

    Step-by-step explanation:

    The Empirical Rule states that, for a normally distributed random variable:

    68% of the measures are within 1 standard deviation of the mean.

    95% of the measures are within 2 standard deviation of the mean.

    99.7% of the measures are within 3 standard deviations of the mean.

    In this problem, we have that:

    Mean = 175cm

    Standard deviation = 6 cm

    Percentage of students below 163 cm

    163 = 175 - 2*6

    So 163 is two standard deviations below the mean.

    By the Empirical rule, 95% of the heights are within 2 standard deviations of the mean. The other 100-95 = 5% are more than 2 standard deviations of the mean. Since the normal distribution is symmetric, 2.5% of them are more than 2 standard deviations below the mean (so below 163cm) and 2.5% are more than two standard deviations above the mean.

    2.5% of the students have heights less than 163cm.

    Out of 1000

    0.025*1000 = 25

    25 is the answer
  2. 18 November, 23:15
    0
    Step-by-step explanation:

    Since the heights of students at a college are normally distributed, we would apply the formula for normal distribution which is expressed as

    z = (x - µ) / σ

    Where

    x = heights of students

    µ = mean height

    σ = standard deviation

    From the information given,

    µ = 175 cm

    σ = 6 cm

    The probability that the height of a student is less than 163 cm is expressed as

    P (x < 163)

    For x = 163

    z = (163 - 175) / 6 = - 2

    Looking at the normal distribution table, the probability corresponding to the z score is 0.023

    Therefore, the expected number of students with heights lesser than 163 cm is

    1000 * 0.023 = 23 students
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