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Chapter 4: Q131E (page 271)

4.131 Chemical composition of gold artifacts. The Journal of Open Archaeology Data(Vol. 1, 2012) provided data onthe chemical composition of more than 200 pre-Columbiangold and gold-alloy artifacts recovered in the archaeologicalregion inhabited by the Muisca of Colombia (a.d.600–1800). One of many variables measured was the percentageof copper present in the gold artifacts. Summary statisticsfor this variable follow: mean = 29.94%, median = 19.75%,standard deviation = 28.37%. Demonstrate why the probabilitydistribution for the percentage of copper in thesegold artifacts cannot be normally distributed.

Short Answer

Expert verified

So here mean and median are not the same, hence the probability distribution cannot have a normal distribution.

Step by step solution

01

Given information

One of many variables measured was the percentageof copper present in the gold artifacts. Summary statisticsfor this variable follow: mean = 29.94%, median = 19.75%,standard deviation = 28.37%.

02

Explanation

The property of normal distribution is mean=median

Here the mean is 29.94% and the median is 28.37%.

So here mean and median are not the same, and therefore the probability distribution cannot have a normal distribution.

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Most popular questions from this chapter

Find the area under the standard normal probability distribution between the following pairs of z-scores:

a)z=0andz=2.00

b)z=0andz=3

c)z=0andz=1.5

d)z=0andz=.80

4.112 California’s electoral college votes. During a presidential election, each state is allotted a different number of votes in the Electoral College, depending on the population. For example, California is allotted 55 votes (the most) while several states (including the District of Columbia) are allotted 3 votes each (the least). When a presidential candidate wins the popular vote in a state, the candidate wins all the Electoral College votes in that state. To become president, a candidate must win 270 of the total of 538 votes in the Electoral College. Chance(Winter 2010) demonstrated the impact on the presidential election of winning California. Assuming a candidate wins California’s 55 votes, the number of additional Electoral College votes the candidate will win can be approximated by a normal distribution with μ=241.5votes and σ=49.8votes. If a presidential candidate wins the popular vote in California, what are the chances that he or she becomes the next U.S. president?

Safety of underground tunnels. Research published in the journal Tunnelling and Underground Space Technology (July 2014) evaluated the safety of underground tunnels built in rigid soils. A factor of safety (FS), measured as the ratio of capacity over demand, was determined for three different areas of tunnels made from shotcrete: tunnel face, tunnel walls, and tunnel crown. FS was determined to be normally distributed in each area, with means and standard deviations shown in the table. Tunnel failure is considered to occur when FS is lower than or equal to 1. Which tunnel area is more likely to result in failure? Why?


Mean

Standard Deviation

Tunnel Face

1.2

0.16

Tunnel Walls

1.4

0.2

Tunnel Crown

2.1

0.7

Traffic fatalities and sporting events. The relationship betweenclose sporting events and game-day traffic fatalities was investigated in the Journal of Consumer Research (December 2011). The researchers found that closer football and basketball games are associated with more traffic fatalities. The methodology used by the researchers involvedmodeling the traffic fatality count for a particular game as a Poisson random variable. For games played at the winner’s location (home court or home field), the mean number of traffic fatalities was .5. Use this information to find the probability that at least three game-day traffic fatalities will occur at the winning team’s location.

4.110 Manufacturing hourly pay rate. Government data indicate that the mean hourly wage for manufacturing workers in the United States is \(20.10 (Bureau of Labor Statistics, January 2016). Suppose the distribution of manufacturing wage rates nationwide can be approximated by a normal distribution with a standard deviation \)1.25 per hour. The first manufacturing firm contacted by a particular worker seeking a new job pays \(21.40 per hour.

a. If the worker were to undertake a nationwide job search, approximately what proportion of the wage rates would be greater than \)21.40 per hour?

b. If the worker were to randomly select a U.S. manufacturing firm, what is the probability the firm would pay more than $21.40 per hour?

c. The population median, call it η, of a continuous random

variable xis the value such that P(x≥η)=P(x≤η)=0.5that is, the median is the value such that half the area under the probability distribution lies above and half lies below it. Find the median of the random variable corresponding to the wage rate and compare it with the mean wage rate.
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