Question

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

Answer 1

Answer:

97.5%

Explanation:

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

1. 68% of data are within μ - σ and μ + σ
2. 95% of data are within μ - 2σ and μ + 2σ
3. 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%