In this chapter we have discussed what a density curve is and specifically focused on a special density curve called the normal distribution. The two critical elements that are necessary for analysis of a density curve are the mean and standard deviation. The mean is the center of the distribution while standard deviation is a measure of spread. We have focused on several key concepts including the 68-95-99.7 rule and z-scores. We then introduced the Normal Distribution Table and the NormalCdf and InvNorm commands to help us be able to move back and forth between probabilities and percentiles and specific values in our distributions.
Chapter 7 Review Exercises
1) Suppose a teacher gives a test in which the scores on the test are normally distributed with a mean of 10 points and a standard deviation of 2 points.
a) Draw a normal curve to represent this situation. Clearly mark the mean and 1, 2, and 3 standard deviations above and below the mean.
b) Using the 68-95-99.7 rule, approximately what percent of students will get a score between 6 and 14?
c) Using the 68-95-99.7 rule, approximately what percent of students will get a score between 8 and 16?
d) Find the percent of students that will get a score between 8 points and 13 points on this test.
e) What percent of students will score at least an 11 points on this test?
f) What percent of students will score between 5 points and 12 points on this test?
g) How many points would a student have to score in order to be at the 90th percentile on this test?
h) What is the z-score associated with a test score of 13 points?
i) How many points did a student score if their z-score was -1.5?
2) Which situation below is most likely to be normally distributed?
i) The heights of all the trees in a forest.
ii) The distances that all the kids at Blaine High School can hit a golf ball.
iii) The number of siblings that each student at Anoka High School has.
iv) The length of time that 6th grade boys at Roosevelt Middle School can hold their breath.
3) The weights of adult male African elephants are normally distributed with a mean weight of 11,000 pounds and standard deviation of 900 pounds.
a) Between what two weights do the middle 50% of all adult male African elephants weigh?
b) Suppose one of these elephants weighs 13,400 pounds. At what percentile is this weight?
c) At what weight would we find the 70th percentile of weights for these elephants?
4) Suppose that IQ test scores are normally distributed with a mean of 100 and a standard deviation of 15.
a) What z-score is associated with and IQ score of 125?
b) The intelligence organization MENSA requires that members score in the top 2.5% of all IQ test takers to gain membership in the organization. What IQ score must a person score to qualify for MENSA?
c) What percentage of IQ scores are greater than 125?
d) What percentage of IQ scores are less than 70? Use the 68-95-99.7 rule to approximate your answer.
e) Who did better, a person with an IQ score of 143 or someone who was at the 99th percentile on the IQ test? Justify your answer.
5) In a certain city, the number of pounds of newspaper recycled each month by a household produces a normal distribution with a mean of 8.5 pounds and a standard deviation of 2.7 pounds.
a) Draw a sketch for this normal distribution and shade in the region that represents the households that recycle between 6 and 12 pounds of newspaper each month.
b) What percent of households recycle between 6 and 12 pounds of newspaper each month?
c) A local newspaper wants to do a story on newspaper recycling in the city. They decide that they would like to base their story on a typical household. After some thought, they decide that 'typical' means that they are in the middle 60% of all households in terms of newspaper recycling. Between what two weights are the 'typical' households?
6) Snowfall each winter in the Twin Cities is normally distributed with a mean of 56 inches and a standard deviation of 11 inches.
a) In what percentage of years does the Twin Cities get less than 3 feet of snow?
b) In what percentage of years does the Twin Cities get more than 6 feet of snow?
c) The winter of 2010-2011 was the fifth snowiest on record for the Twin Cities with a total snowfall of 85 inches. What percentage of years will have snowfalls of more than 85 inches?
d) A winter is considered to be dry if it in the lowest 10% of snowfall totals. What is the maximum amount of snow the Twin Cities could receive to still be called a dry winter?
7) You just got your history test back and found out you scored 37 points. The scores were normally distributed with a mean of 31 points and a standard deviation of 4 points. When you tell your parents how you did, your little brother pipes in that he got a 56 on his math test which was normally distributed with a mean of 40 points and a standard deviation of 11 points. How could you use z-scores to explain to your parents that your score was more impressive than your little brother's score?
8) In 1941, Ted Williams batted 0.406 for the baseball season. He is the last player to hit over 0.400 for an entire major league baseball season. In 2009, Joe Mauer hit 0.365 for the baseball season. In 1941, the batting averages were normally distributed with a mean of 0.260 and a standard deviation of 0.041. In 2009, the batting averages were normally distributed with a mean of 0.262 and a standard deviation of 0.035. Decide which player had a better season compared to the rest of the league during their respective year by comparing z-scores.
9) Suppose that medals will be given out to any student at Andover High School that scores at least 200 points on an aptitude test. The mean score on the aptitude test is 150 points with a standard deviation of 22 points. How many medals should be ordered if there are 456 students who sign up for the test?
Density Curve www.madscientist.blogspot.com
Skewed Distributions http://en.wikipedia.org/wiki/Skewness
68-95-99.7 Normal Curve www.rahulgladwin.com
Pet Store Window www.teddyhilton.com
Traffic Jam www.rnw.nl
Leafcutter Ant www.orkin.com/ants
Pair of Queens http://www.123rf.com
American Diabetes Association http://americandiabetesassn.wordpress.com
Track Race http://www.tierraunica.com
Joe Mauer http://www.mauersquickswing.com