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Z-scores and Empirical Rule Notes Name: _______________________ Empirical Rule says that … 68% of the data in a normally distributed data set is within 1 standard deviation. 95% of the data in a normally distributed data set is within 2 standard deviations. 99.7% of the data in a normally distributed data set is within 3 standard deviations. Types of questions involving the Empirical Rule: 1. The scores on a university examination are normally distributed with a mean of 70 and a standard deviation of 10. If the middle 68% of students will get a “C”, what is the lowest mark that a student can have and still be awarded a C? To solve: The middle 68% of students are within 1 standard deviation of the mean according to the Empirical Rule. The question wants to know the LOWEST mark that a student can get o receive a C, so you must subtract 1 standard deviation from the mean. 70 – 10= 60 2. The lifetime of lightbulbs of a particular type are normally distributed with a mean of 100 mmHg and a standard deviation of 6 mmHg. What percentage of 18-year-old women have a systolic blood pressure between 88 mmHg and 112 mmHg? To solve: You must decide how many standard deviations away each of the given blood pressures are from the mean. Start by looking at the left of the mean. 100 – 6 = 94. That’s not far enough, so you subtract another standard deviation. 94 – 6 = 88. Because 88 is two standard deviations away from the mean, 95% of 18-year-old women have a systolic blood pressure between 88 mmHg and 112 mmHg. Z-scores: xx Z-scores are used to normalize data, or to convert all data to a common unit. A z-score tells you how many standard deviations away your data is from the mean. Types of z-score questions: 1. Lewis earned 80 on his biology midterm and a 71 on his history midterm. In the biology class the mean score was 75 with a standard deviation of 4. In the history class the mean score was 73 with a standard deviation of 3. a. Convert each score to a standard z-score. To solve: Biology: 80 75 1.25 4 History: 71 73 0.67 3 b. On which test did he do better compared to the rest of the class? Solution: In both classes he did worse than the average because both z-scores were negative. However, he did better compared to the rest of the class in History because his z score is smaller. Z-table: The z-table gives you the area, probably, or percent of data that is below the said value. You use the z-table when you see one of the above bold words. How to use the z-table: You need to have one number before the decimal, and two numbers after the decimal. If there is no number before the decimal, put a 0 before the decimal. If there is only one number after the decimal, add a 0 on the end. If there is no decimal, add one then add two 0s after it. If there are more than two numbers after the decimal you have to ROUND. o Look at the 4th number after the decimal. If it is a 4 or below, keep the 3rd number the same. Example: 1.45345698 becomes 1.45 because 3 is the 4th number. o Look at the 4th number after the decimal. If it is a 5 or above, raise the 3rd number 1 higher than it was before. Example: -0.86795643 becomes -0.87 because 7 is the 4th number. Types of z-table questions: A class of 217 students participated in a softball throw for the distance test. The mean performance of the group was 173 and the standard deviation was 31. Based on this data, answer the following questions: a. What percentage of students was able to throw the softball between 151 and 180? To solve: Because the questions asks for the percentage, you must use your z-table. In order to use your z-table, you must convert your throw values to z-scores. 151 173 180 173 0.71 when rounded 0.23 when rounded 31 31 Next find both of these z-scores on the z-table: 0.2389 and 0.5910 To find the percentage between two numbers you subtract the lower from the higher: 0.5910 – 0.2389 = 0.3521 = 35.21% b. What percentage of the students could throw farther than 200 feet? 200 173 0.87 . Then find the z-table value: 0.8078 To solve: Find the z-score first: 31 To find the percentage of students who throw farther than 200, you must subtract your z-table value from 1. 1 – 0.8078 = 0.1922 = 19.22% c. What percentage of the students could only throw less than 114 feet? To solve: The z-table values give you the percent that throws less than the given amount. Therefore, once you find the z-score, simply look it up on your z-table. 114 173 1.90 Then find the z-table value: 0.0287 = 2.87% Z-score: 31 Normal Distribution: A normal distribution looks like a bell curve. In order to use the Empirical Rule or a Z-table your data must be normally distributed. Use what you know about a normal distribution to answer the following questions: 1. Which graph above has a larger mean? Solution: Graph B has a larger mean because the mean is located in the middle of each graph and the mean of graph B is located further to the right. 2. Which graph has a larger standard deviation? Solution: Graph B has a larger standard deviation because it is more spread apart. Exercises: 1. What percent of data is within 1 std deviation of the mean? 2. What percent of data is within 2 std deviations of mean? 3. What percent of data is within 3 std deviations of mean? 4. The scores on a university examination are normally distributed with a mean of 62 and a standard deviation of 11. If the middle 68% of students will get a “C”, what is the lowest mark that a student can have and still be awarded a C? 5. The lifetimes of lightbulbs of a particular type are normally distributed with a mean of 360 hours and a standard deviation of 5 hours. What percentage of the bulbs have lifetimes that lie within 2 standard deviation of the mean? A) 31% B) 84% C) 68% D) 95% 6. The systolic blood pressure of 18-year-old women is normally distributed with a mean of 120 mmHg and a standard deviation of 12 mmHg. What percentage of 18-year-old women have a systolic blood pressure between 96 mmHg and 144 mmHg? A) 99% B) 68% C) 95% D) 99.99% 7. Lewis earned 85 on his biology midterm and 81 on his history midterm. In the biology class the mean score was 79 with a standard deviation of 5. In the history class the mean score was 76 with a standard deviation of 3. (a) Convert each score to a standard z score. (b) On which test did he do better compared to the rest of the class? 4. On one measure of attractiveness, scores are normally distributed with a mean of 3.93 and a standard deviation of .75. Find the probability of randomly selecting a subject with a measure of attractiveness that is greater than 2.75. 5. The serum cholesterol levels in men aged 18 to 24 are normally distributed with a mean of 178.1 and a standard deviation of 40.7. If a man aged 18 to 24 is randomly selected, find the probability that his serum cholesterol level is between 100 and 200.