Question

1. According to the empirical rule, for a distribution that is symmetric and bell-shaped, approximately _______ of the data values will lie within 3 standard deviations on each side of the mean.

2. Assuming that the heights of boys in a high-school basket- ball tournament are normally distributed with mean 70 inches and standard deviation 2.5 inches, how many boys in a group of 40 are expected to be taller than 75 inches?

3. Let x be a random variable that represents the length of time it takes a student to complete Dr. Gill’s chemistry lab project. From long experience, it is known that x has a normal distribution with mean μ = 3.6 hours and standard deviation σ = 0.5.

Convert each of the following x intervals to standard z intervals.

(a) x ≥ 4.5 4. (a) __________________________

(b) 3 ≤ x ≤ 4 (b) __________________________

(c) x ≤ 2.5 (c) __________________________

Convert each of the following z intervals to raw-score x intervals.

(d) z ≤ −1 (d) __________________________

(e) 1 ≤ z ≤ 2 (e) __________________________

(f) z ≥ 1.5 (f)_ __________________________

Answer 1

Empirical rule :

The empirical rule states that for a normal distribution, nearly all of the data will fall within three standard deviations of the mean. The empirical rule can be broken down into three parts:

• 68% of data falls within the first standard deviation from the mean.
• 95% fall within two standard deviations.
• 99.7% fall within three standard deviations.

1. According to the empirical rule, for a distribution that is symmetric and bell-shaped, approximately 99.7% of the data values will lie within 3 standard deviations on each side of the mean.

2. Assuming that the heights of boys in a high-school basket- ball tournament are normally distributed with mean 70 inches and standard deviation 2.5 inches, how many boys in a group of 40 are expected to be taller than 75 inches

X : Height of boys in a high school basket- ball tournament

X follows normal distribution with mean 70 inches and standard deviation 2.5 inches.

Probability that a randomly selected boy taller than 75 inches = P(X>75)

75 = 70 + 2 x 2.5 i.e mean + 2 standard deviations;

From the The empirical rule for a normal distribution:

95% fall within two standard deviations i.e between (mean - 2 standard deviation) and (mean + 2 standard deviation). i.e

For given problem ,

P(70-2x2.5 < X < 70+2x2.5) = P(65 < X < 75) = 0.95

i.e P(X>75) + P(X<65) = 1-0.95=0.05

As normal distribution is symmetric,

P(X>75) = 0.05/2=0.025

P(X<65) =0.05/2=0.025

Probability that a randomly selected boy taller than 75 inches = P(X>75) =0.025

Number of boys in a group of 40 are expected to be taller than 75 inches = 40 x Probability that a randomly selected boy taller than 75 inches 40 x 0.025= 1

Number of boys in a group of 40 are expected to be taller than 75 inches = 1

x has a normal distribution with mean = 3.6 hours and standard deviation = 0.5.

Z-score = (x-mean)/Standard deviation

Convert each of the following x intervals to standard z intervals.

(a) x ≥ 4.5 4. (a) Z -score for 4.5 = (4.5 - 3.6)/0.5 = 0.9/0.5 = 1.8; z interval : z 1.8

(b) 3 ≤ x ≤ 4 (b) Z-score for 3 = (3 - 3.6)/0.5 = -0.9/0.5 = -1.2; Z-score for 4 = (4 - 3.6)/0.5 = 0.4/0.5 = 0.8 z interval : -1.2z 0.8

(c) x ≤ 2.5 (c) Z-score for 2.5 = (2.5 - 3.6)/0.5 = -1.1/0.5 = -2.2; z interval = -2.2

Convert each of the following z intervals to raw-score x intervals.

Z-score = (x-mean)/Standard deviation

raw score x = mean + z x standard deviation

(d) z ≤ −1 (d) x for 1= 3.6 +(-1) x 0.5 = 3.6-0.5 = 3.1; raw-sore x interval : x 3.1

(e) 1 ≤ z ≤ 2 (e) x for 1 = 3.6 + 1x0.5_= 3.6+0.5 =4.1 ; x for 2 = 3.6 +2x0.5 = 3.6+1 =4.6; raw-score interval : 4.1x4.6

(f) z ≥ 1.5 (f)_ x for 1.5 = 3.6 + 1.5 x 0.5 = 3.6 + 0.75 = 4.35 ; raw-score interval : x4.35