Z-Scores and What Exactly Does It Mean to be Normal Anyway? Discrete Math, Section 9.5

Transcription

1 Z-Scores and What Exactly Does It Mean to be Normal Anyway? Discrete Math, Section 9.5 We first look at the concept of a z-score in isolation, then we tie it into the standard normal distribution. I. Z-Scores Another way to compare individual data points to the mean of a set of data is to measure how far these points lie from the mean. This time, instead of using the units of measurement of the variable, we use standard deviations as the unit. A z-score, computed as follows, measures this distance: x! x z = s As with the standard deviation formulas, the above formula is for a sample. If the population standard deviation is known, we use this formula: z = x " µ # Example 1: Z-Scores Consider the set of numbers {1, 3, 6, 7, 8}. a. Calculate the mean and standard deviation of this data. b. Calculate the z-score for each number in the original data set. c. What do you notice about the z-scores of numbers that are greater than the mean? d. What do you notice about z-scores of numbers which are less than the mean? e. What does it mean when the z-score of a number is 0? f. Calculate the mean of the z-scores. g. Calculate the standard deviation of the z-scores. (Use the calculator!) Notes of importance: 1. The sample mean of the z-scores is. 2. The sample standard deviation of the z-scores is. Page 1

2 Example 2 (BPS): SAT Scores Eleanor scores 680 on the math part of the SAT. The mean score of this section of the SAT is 500 with a standard deviation of 100. Gerald takes the ACT mathematics test and scores 27. The mathematics section the ACT has a mean of 18 and a standard deviation of 6. Find the standardized scores for both students. Assuming that both tests measure the same type of ability, who has the higher score? II. Density Curves and The Normal Distribution Definition (BPS) : A density curve is a curve that 1. always lies on or above the horizontal axis 2. has area exactly 1 underneath it. We'll see that this isn't a terribly different concept from what we have been doing Examples (1) Symmetric density curve Mean = Median (2) Right-skewed density curve Mean > Median (3) Left-skewed density curve Mean < Median Notes of importance: 1. The median of a density curve is the point that divides the area under the curve in half. 2. The mean of a density curve is the point at which the curve would balance if it were made of solid material. Page 2

3 3. A density curve approximates the shape of an actual distribution. It is used to approximate the proportion of observations that fall within a certain range of values. For example, if we were to collect the heights of 100,000 adult men in the United States, a graph we could use to display this data is a histogram. The shape of the histogram would probably be similar to a bell, and the perfect bell-shaped curve which approximates the shape of this histogram is the density curve associated with this question. The most commonly used family of density curves is the family of normal curves, which are graphs of the normal distributions. These graphs are symmetric and bell-shaped, with mean and standard deviation determining the center and spread of the curve. Example of a normal distribution (with mean and standard deviation labeled): Recall the Rule from the last section. What did it say? We now firm up that definition a bit. For a perfectly normal distribution, this rule is always true. Example 3 ( Understandable Statistics): The incubation time for Rhode Island Red chicks is normally distributed with mean of 21 days and standard deviation of approximately 1 day (based on information from World Book Encyclopedia). Use the Rule to answer the following questions. (a) What percent of chicks do we expect will hatch in days? (b) What percent of chicks do expect will hatch in days? (c) What percent of chicks do we expect will hatch in 21 days or fewer? (d) What percent of chicks do we expect will hatch in 18 to 24 days? Page 3

4 Example 4 (Understandable Statistics): The playing life of a Sunshine Radio is normally distributed with mean 600 hours and standard deviation 100 hours. What is the probability that a radio selected at random will last... (a) from 600 to 700 hours? (b) more than 800 hours? (c) less than 300 hours? Question: Suppose we are asked for the proportion of values under a normal curve that are not exactly 1, 2, or 3 standard deviations from the mean. How can these values be calculated? Answer: To find these proportions, we need to find areas under portions of normal curves. These areas will be tied into the idea of probability. There are two ways to find such areas: Method #1: Using a standard normal distribution table Method #2: Using a TI graphing calculator We ll discuss method 1 first. Since there are an infinite number of normal distributions, we need some way of converting all normal distributions into one standard distribution, known as the standard normal distribution. The standard normal distribution is a normal distribution with mean 0 and standard deviation 1. The standard normal distribution is represented by a density curve known as the standard normal curve. A table of areas under the standard normal curve can be found in your text. Example 5: Practice using the standard normal distribution table Find the standard normal curve area that lies (a) to the left of z = 0.85 (b) to the left of z = (c) to the right of z = 1.23 (d) to the right of z = (e) between z = 1.12 and z = 2.79 (f) between z = and z = 0.44 Page 4

5 Before using this table to solve problems involving normal distributions, however, we must convert (standardize) values from normal distributions to standard normal values. To do so, we use z-scores! In most problems, we'll know the mean and standard deviation of the population, so we use this formula: z = x " µ # Notes of importance: 1. When we don't know the mean and standard deviation of the population, we approximate using sample statistics. (The formula looks the same just the names of variables change.) 2. A z-score measures the number of standard deviations between an observation x and the mean µ of the data set. 3. The area under a normal curve between two given values is the same as the area under the standard normal curve between their standardized scores (z-scores). Example 6 (Understandable Statistics) A healthy 10-week-old (domestic) kitten should weigh an average of 24.5 ounces with a standard deviation of 5 ounces. (a) What is the probability that a healthy 10-week-old kitten will weigh less than 14 ounces? (b) What is the probability that a healthy 10-week-old kitten will weigh more than 33 ounces? (c) What is the probability that a healthy 10-week-old kitten will weigh between 14 and 33 ounces? (d) A kitten whose weight is in the bottom 10% of the probability distribution of weights is called undernourished. What is the cutoff point for the weight of an undernourished kitten? Example 7 (Freedman) In a law school class, the entering students averaged about 30 on the LSAT with a standard deviation of about 8. The histogram of LSAT scores followed the normal curve reasonably well. (a) About what percentage of the class scored below 36? (b) One student was 0.5 standard deviations above average on the LSAT. About what percentage of the students had lower scores than she did? Page 5

6 Example 8 (SDA) An economic forecaster suggests that interest rates will be 9% next year and that the uncertainty in the forecast is indicated by a standard deviation of 2 percentage points. Assume that the actual interest rate is normally distributed with this mean and standard deviation. (a) Find the probability that the forecaster is correct to within 1 percentage point on either side of the 9% forecasted. (b) Find the probability that interest rates will be 8% or smaller. Example 9 (Understandable Statistics) Quick Start Company makes 12-volt batteries. After many years of product testing, the company knows that the average life of a Quick Start battery is normally distributed, with a mean of 45 months and a standard deviation of 8 months. (a) If Quick Start guarantees a full refund on any battery that fails within the 36 month period after purchase, what percentage of its batteries will the company expect to replace? (b) If Quick Start does not want to replace more than 10% of its batteries under the full-refund guarantee policy, for how long should the company guarantee the batteries (to the nearest month)? On to the calculators All of the areas under normal curves can be found using the TI graphing calculators (this is method 2 that we discussed earlier). The two functions you want to know are normalcdf (which calculates the areas under normal curves) and invnorm (which calculates the value of a normal distribution which has a specified area to the left of it). The key sequences are: normalcdf(left endpoint, right endpoint, mean, standard deviation) invnorm(area to the left of the point in question, mean, standard deviation) It is much easier to use the graphing calculator to compute these values since no conversions (involving z -scores), and therefore, no tables, are necessary. Obviously, it will be important to know when to use which calculator feature. normalcdf calculates areas (which represent percentages, proportions or probabilities) and invnorm calculates actual values from a normal distribution. Here s an example illustrating the difference from Statistics in Action. Page 6

7 Example 10 (Statistics in Action) Which of the following situations are unknown percentage problems and which are unknown value problems? For each, draw and label a normal curve, showing the three quantities that are given and the one quantity to find. (a) In a recent year, students entering the University of Florida had a mean SAT I score of 1135, with standard deviation 180. The distribution was roughly normal. What percentage of SAT I scores were greater than 1300? (b) In 2000, the mean SAT I math score nationally was 514, with a standard deviation of 113. Find the upper quartile of the distribution. We'll continue our look at statistics with an application that uses the standard normal distribution. That's the next set of notes though Page 7