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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.渭卤蟽b.渭卤2蟽c.渭卤3蟽

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,

P渭-蟽<X<渭+蟽=P渭-蟽-渭蟽<X-渭蟽<渭-蟽-渭蟽=P-1<Z<1=PZ<1-PZ鈮�-1=0.84134-0.15866=0.68269饾啅0.68P渭-蟽<X<渭+蟽饾啅0.68

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

03

Calculating the proportion when

P渭-2蟽<X<渭+2蟽=P渭-2蟽-渭蟽<X-渭蟽<渭-2蟽-渭蟽=P-2<Z<2=PZ<2-PZ鈮�-2=0.97725-0.02275=0.9545饾啅0.95P渭-2蟽<X<渭+2蟽饾啅0.95

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

04

Calculating the proportion when

c.

P渭-3蟽<X<渭+3蟽=P渭-3蟽-渭蟽<X-渭蟽<渭-3蟽-渭蟽=P-3<Z<3=PZ<3-PZ鈮�-3=0.99865-0.00135=0.9973饾啅1P渭-3蟽<X<渭+3蟽饾啅1

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

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Most popular questions from this chapter

Establishing tolerance limits. The tolerance limits for a product's quality characteristic (e.g., length, weight, or strength) are the minimum or maximum values at which the product will operate properly. Tolerance limits are set by the engineering design function of the manufacturing operation (Total Quality Management, Vol. 11, 2000). The tensile strength of a particular metal part can be characterized as being normally distributed with a mean of 25 pounds and a standard deviation of 2 pounds. The part's upper and lower tolerance limits are 30 pounds and 21 pounds, respectively. A part that falls within the tolerance limits results in a profit of \(10. A part that falls below the lower tolerance limit costs the company \)2; a part that falls above the upper tolerance limit costs the company $1. Find the company鈥檚 expected profit per metal part produced.

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Soft-drink dispenser. The manager of a local soft-drink bottling company believes that when a new beverage dispensing machine is set to dispense 7 ounces, it in fact dispenses an amount at random anywhere between 6.5and 7.5 ounces inclusive. Suppose has a uniform probability

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