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Chapter 4: Q117E (page 268)

If a population data set is normally distributed, what isthe proportion of measurements you would expect to fallwithin the following intervals?

a.μ±σb.μ±2σc.μ±3σ

Short Answer

Expert verified

a. The proportion of measurements that fall in the interval is 68%

b.The proportion of measurements that fall in the interval is 95%

c.The proportion of measurements that fall in the interval is 100%

Step by step solution

01

Given Information

The population is normally distributed.

02

Calculating the proportion when μ±2σ

b.

From the property of normal distribution,

Pμ-σ<X<μ+σ=Pμ-σ-μσ<X-μσ<μ-σ-μσ=P-1<Z<1=PZ<1-PZ≤-1=0.84134-0.15866=0.68269𝆏0.68Pμ-σ<X<μ+σ𝆏0.68

The proportion of measurements that fall within 1 standard deviation of the mean is approximately 68%.

03

Calculating the proportion when

Pμ-2σ<X<μ+2σ=Pμ-2σ-μσ<X-μσ<μ-2σ-μσ=P-2<Z<2=PZ<2-PZ≤-2=0.97725-0.02275=0.9545𝆏0.95Pμ-2σ<X<μ+2σ𝆏0.95

The proportion of measurements that fall within 2 standard deviations of the mean is approximately95%.

04

Calculating the proportion when

c.

Pμ-3σ<X<μ+3σ=Pμ-3σ-μσ<X-μσ<μ-3σ-μσ=P-3<Z<3=PZ<3-PZ≤-3=0.99865-0.00135=0.9973𝆏1Pμ-3σ<X<μ+3σ𝆏1

The proportion of measurements that fall within 3 standard deviations of the mean is approximately 100%.

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