Review 5 Normal Distributions & Z-Scores Test 5 will be written on Monday May 26 th during Block E in 2603. 1. Which shape of distribution best describes a normal distribution? 2. List 5 properties of a normal distribution. 3. State the standard deviation and mean for each of the following. (a) X~N(10, 32) (b) X~N( 2, 4), 4. Find he z-score corresponding to the 44th percentile. 5. Given a normally-distributed data set whose mean is 40 and whose standard deviation is 8, what value of x would have a z-score of 1.25? 6. The volume of cola in 355-mL cans is normally distributed with a standard deviation of 3 mL. What percent of cans have a volume greater than 359 mL? 7. If X~N(40, 32), what percent of the data are between 36 and 41? 8. Find the percentile corresponding to x = 15 if X~N(12, 2.62). 9. The masses of bolts made in a plant are normally distributed. Bolts will be rejected if their z-scores are greater than 2.15 or less than 2.10. What percent of bolts will be rejected? 10. The time that a certain top sprinter takes to run the 100-m dash is normally distributed with a mean of 9.8 s and a standard deviation of 0.2 s. In what percent of his sprints will his time be less than 10.2 s? Include a diagram. 11. A lake in Ontario has a mean June water temperature of 20.3°C and a standard deviation of 2.1°C. How many days in June (30 days) would you expect to find comfortable for swimming if you will only swim when the water temperature is at least 21.5°C? Show all work. 12. Ravi’s math contest result put him in the 97th percentile. If 4000 students competed, how many had a score higher than Ravi’s score? 13. What normal z-score corresponds to the 14th percentile? 14. Suppose X~N(50, 16). What value of x would have a z-score of 2.10? 15. In X~N(12, 22), what percent of the data is between 9 and 13? Include a diagram. 16. Explain why the data set below does not resemble a normal distribution: 0, 0, 0, 2, 2, 4, 6, 6, 8, 8 8. Show your work. 17. Explain what information is given by the magnitude and sign of the z-score of a piece of data. 18. How many of the 250 perch netted by a fisherman would you expect to have a mass more than 400 g, if the distribution of their masses in approximately normal with a mean of 300 g and a standard deviation of 80 g? 19. For X~N( , place. ), approximately 48% of the data are less than 2.9. Find the value of , to one decimal 20. Why do normal z-score tables typically not include z values greater than 2.99? 21. Given the following residual plot, state why you would use a curve of best fit instead of a least squares line. Scatter Plot Collection 1 20 y 16 12 8 4 Resi... 0 0 2 4 6 x 8 10 12 0 2 4 6 x 8 10 12 2 -4 y = 1.02x + 0.49; r^2 = 0.55 22. (a) Using the linear regression line y  1.02 x  0.49 from question 22 above, calculate the residual for the point (5,  3.1) . Show your work. (b) What conclusion can you make when the residual value is negative? Formulas x x n w x  w i i xw xw i  2 s N  R1  y1  a x1   b  i z i i i x   fm  f i i i x  x  n 1 2 s  fi  mi  x  n 1 2 xx s