MTH 125 Lab 6 Normal Distributions and Empirical Rule Name ___________________________ For a certain law school, the entering students have an average LSAT score of about 700 with a standard deviation of about 40. The smoothed density histogram of the LSAT scores appears to follow a normal distribution. Sketch a smoothed density histogram (a bell-shaped curve) of the distributions for the LSAT scores. (For this problem, draw this on your own paper. You will not need to submit this graph.) 620 660 700 740 780 Because this distribution is approximately normal, the mean is equal to the median. Therefore, the mean will also divide our data into two equal pieces. We will use the mean as our measure of center. 1. Locate the mean on the distribution and mark it (Again, do this on your own paper.) 2. What fraction of the data is to the left of the mean? How much is to the right of the mean? 3. Calculate the z (standardized) score for an LSAT score of 700. Write that value here. (You may know this without any calculations) 4. Find the interval of LSAT scores that is 1 standard deviation away from the mean. Write the interval here and mark it on your smoothed histogram. MTH 125 Lab 6 5. Using the lower value of your interval, calculate the z (standardized) score. Do the same for the upper value of your interval. What did you notice? 6. Using the z-scores you just found in #6, find the area under the curve. Change it to a percentage. 7. Find the interval of LSAT scores that is 2 standard deviations away from the mean. Write the interval here and mark it on your smoothed histogram. 8. Using the lower value of your interval, calculate the z (standardized) score. Do the same for the upper value of your interval. What did you notice? 9. Using the z-scores you just found in #9, area under the curve. Change it to a percentage. 10. Find the interval of LSAT scores that is 3 standard deviations away from the mean. Write the interval here and mark it on your smoothed histogram. 11. Using the lower value of your interval, calculate the z (standardized) score. Do the same for the upper value of your interval. What did you notice? 12. Using the z-scores you just found in #12, find the area under the curve. Change it to a percentage. MTH 125 Lab 6 According to the Empirical Rule, about 68% of the data will be located within 1 standard deviation from the mean (from -1 to +1). These are also the z-scores. About 95% of the data is located within 2 standard deviations from the mean (from -2 to +2) and about 99.7% of the data is located within 3 standard deviations from the mean (from -3 to +3). Hopefully, the percentages you calculated match these values. These standardized values allow us to determine how far away from the mean a particular data point is located. 13. If 68% of the data is within 1 standard deviation of the mean (from -1 to +1), what percentage of the LSAT scores is between a z-score of 0 and +1? Between a z-score of -1 and 0? Write these values as decimals. 14. Find the area under the curve for a z-score of +1 and subtract the area under the curve for a z-score of 0. This value, if rounded to 2 decimal places, should look very familiar. (Hint: Does it match the percentage from #14?) If we continue examining the relationship between the number of standard deviations from the mean, the z-scores, and the area under the curve we will find there is a direct correlation between these values. In the meantime, we will practice calculating z-scores. 15. For an LSAT score of 750, find the z-score. How many standard deviations away from the mean is this score? 17. What percentage of the students scored below 750? 18. If there are 9000 students entering law school, how many students scored below 750 on the LSAT? 19. Joe's last LSAT score was 0.5 standard deviations below the average LSAT score. Approximately what percent of entering students have an LSAT score lower than Joe's? MTH 125 Lab 6 20. Approximately what percent of the entering students have an LSAT score higher than Joe's? How did you find this value? 21. Of the 9000 entering students, how many students scored higher than Joe? 22. Find the LSAT score that has a z-score of -1.6. 23. Find the percent of students who scored lower than the score you found in #22. 24. This law school has a program to help entering students who have low LSAT scores. The students with an LSAT score in the lowest 15% of the distribution are required to attend this program. What is the minimum score needed for a student to be exempt from this requirement? 25. How many of the 9000 entering students will be required to attend this program? 26. This particular law school is offering scholarships to those entering students who score in the top 1%. Find the LSAT score that those students would need to obtain to be eligible for the scholarship. MTH 125 Lab 6