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29 November, 00:54

Suppose that diameters of a new species of apple have a bell-shaped distribution with a mean of 7.2cm and a standard deviation of 0.38cm. Using the empirical rule, what percentage of the apples have diameters that are no more than 7.96cm

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  1. 29 November, 02:33
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    97.5%

    Explanation:

    By the empirical rule (68-95-99.7),

    68% of data are within μ - σ and μ + σ 95% of data are within μ - 2σ and μ + 2σ 99.7% of data are within μ - 3σ and μ + 2σ

    σ and μ are the standard deviation and the mean respectively.

    From the question,

    μ = 7.2 cm

    σ = 0.38 cm

    7.96 = 7.2 + (n * 0.38)

    n = 2

    Hence, 7.96 represents μ + 2σ.

    P (X < μ + 2σ) = P (X < μ) + P (μ < X < μ + 2σ)

    P (X < μ) is the percentage less than the mean = 50%.

    P (μ < X < μ + 2σ) is half of P (μ - 2σ < X < μ + 2σ) = 95% : 2 = 47.5%.

    Considering this, for apples that are no more than 7.96 cm,

    P (X < 7.96) = P (X < 7.2) + P (7.2 < X < 7.96) = 50% + 47.5% = 97.5%
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