MATH 110 Week 9 Chapter 17 Worksheet The Mathematics of Normality NAME Normal (bell-shaped) distributions play an important role in the world of statistics. One reason the normal distribution is important is that many psychological and educational variables are distributed approximately normally. A second reason the normal distribution is so important is that it is easy for mathematical statisticians to work with. This means that many kinds of statistical tests that scientists use can be derived for normal distributions. Lots statistical tests assume normal distributions. Fortunately, these tests work very well even if the distribution is only approximately normally distributed. 17.1 Approximately Normal Data Sets Review the Examples on p. 518-521 and compare the bar graphs that appear as approximately bell shaped. approximately normal distribution normal distribution When a large real-life data set has an approximately normal distribution, we can get a lot of useful information about the data set as if it were an idealized normal distribution. Normal distributions share a lot of nice properties and the idea is to take advantage of these properties even when the distribution is only approximately normal. In the next two section, we will discuss how this is done. 17.2 Normal Curves and Normal Distributions What is a normal curve? The study of normal curves can be traced back to the great German mathematician Carl Friedrick Gauss (1777-1855). For this reason, normal curves are sometimes called Gaussian curves. 1
The following is a summary of some of the key mathematical properties of normal curves. 1. Symmetry 2. Median and Mean 3. Standard Deviation 4. Quartiles 5. The 68-95-99.7 Rule Example: Normal Data Set Consider a data set having a normal distribution with mean µ = 497 and standard deviation σ = 114. Use the various properties of a normal distribution of analyze the data. 2
Standardizing Normal Data We have seen that normal curves don t all look alike but this is only a matter of perception. In fact, all normal distributions tell the same underlying story but use slightly different dialects to do so. One way to understand the story of any given normal distribution is to rephrase it in a simple common language a language that uses the mean µ and the standard deviation σ as its only vocabulary. This process is known as standardizing the data. To standardize the data value x, Example: Standardizing Normal Data Consider a normally distributed set with mean µ = 45 ft and standard deviation σ = 10 ft. Standardize the following data values: 1. x 1 = 55 ft 2. x 2 = 35 ft 3. x 3 = 50 ft 4. x 4 = 21.58 ft Standardizing Rule: Examples: 1. Consider a normally distributed data set with mean µ = 63.18 inches and standard deviation σ = 13.27 inches. What is the z-value of x = 91.54 inches? 2. Consider a normal distribution with mean µ = 235.7 grams and standard deviation σ = 41.58 grams. What is the data value x that corresponds to the standardized z-value z = 3.45? 3
17.3 Modeling Approximately Normal Distributions Let s analyze a real life approximately normal data set using the information that we have learned thus far: Example: 2011 SAT Critical Reading Scores Refer back to Example 17.3 for the data involving the 2011 scores in the Critical Reading section of the SAT. Note that the total number of test takers was N = 1, 647, 123 and the mean score was µ = 497 and standard devation was σ = 114. Model the data using a normal distribution. (Keep in mind that SAT section score go in increments of 10 points.) 17.4 Normality in Random Events When you toss a coin 100 times the number of heads is variable meaning, you do not expect to get the exact same number of heads every time you do it. To a large extent, the number of heads depends on the laws of chance. Such a variable is called a random variable. If instead of looking at trials consisting of 100 coin tosses, we generalize to trials consisting of n tosses, we will still get a bell-shaped distribution but the values µ and σ would change. Specifically, Honest-Coin Principle: This means that if we toss n = 100 fair coin tosses, we should be able to predict the mean and standard deviation: What if a fair coin is tossed 156 times? Say that you can make a bet that if the number of heads is between 120 and 136, we win. Should we make this bet? 4
What if the coin is dishonest? Dishonest-Coin Principle: Suppose a coin is rigged so that it comes up heads only 20% of the time and that it is tossed 100 times. What are the approximated mean and standard deviation? Example: The 2012 Presidential Election Consider a poll conducted by the Monmouth University Polling Institute the weekend of November 1-4, 2012 just days before th 2012 election. The poll consisted of a national random sample of 1,417 likely voters who were asked If the election were today, would you vote for Mitt Romney the Republican or Barak Obama the Democrat or some other candidate? The results were 48% for Obama, 48% for Romney, 2% other and 2% undecided. In this example, if we disregard the undecided and other votes, we can think of this poll like a coin toss labeling Romney as heads and Obama as tails. In this case, half of the coins came up heads. What are the approximated mean and standard deviation? Homework: p.536 # 1, 3, 13, 15, 23, 25, 41, 57 5