ࡱ> *,)` #bjbj A4<a"p<MMM!!!!!!!$#h9&"M+"MMM"m"M!M!t  0 Nc !1"0a" ,&i"& &!TMMMMMMM"" MMMa"MMMM\d\ 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. Thats 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:  EMBED Equation.DSMT4  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:  EMBED Equation.DSMT4  History:  EMBED Equation.DSMT4  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. 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. 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.  EMBED Equation.DSMT4  when rounded  EMBED Equation.DSMT4  when rounded 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? To solve: Find the z-score first:  EMBED Equation.DSMT4 . Then find the z-table value: 0.8078 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. Z-score:  EMBED Equation.DSMT4  Then find the z-table value: 0.0287 = 2.87% 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. 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