The price-earnings (PE) ratios of a sample of stocks have a mean value of 10.5 and a standard deviation of 3. If the PE ratios have a normal distribution, use the Empirical Rule (also called the 68-95-99.7 Rule) to estimate the percentage of PE ratios that fall between:

a) 68% falls within 7.5 and 13.5

b) 95% falls within 4.5 and 16.5

c) 99.7% falls within 1.5 and 19.5

Step-by-step explanation:

Given that:

mean (μ) = 10.5, Standard deviation (σ) = 3.

The Empirical Rule (also called the 68-95-99.7 Rule) for a normal distribution states that all data falls within three standard deviations. That is 68% falls within the first standard deviation (µ ± σ), 95% within the first two standard deviations (µ ± 2σ), and 99.7% within the first three standard deviations (µ ± 3σ).

Therefore using the empirical formula:

68% falls within the first standard deviation (µ ± σ) = (10.5 ± 3) = (7.5, 13.5)

95% within the first two standard deviations (µ ± 2σ) = (10.5 ± 2 (3)) = (10.5 ± 6) = (4.5, 16.5)

99.7% within the first three standard deviations (µ ± 3σ) = (10.5 ± 3 (3)) = (10.5 ± 9) = (1.5, 19.5)