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6.4 The Standard Normal Distribution Upon completion of this activity and any associated homework, you should be able to: • Shade in an area under a normal curve that represents the probability of a continuous random variable taking on an interval of values • Use a graphing calculator to find probabilities for a continuous random variable • Use a graphing calculator to find data values given the probability of a continuous random variable taking on an interval of values 1. The height X (in inches) of women in the United States is considered to be normally distributed with a mean of 64 and standard deviation of 2.7. This is denoted N (64, 2.7) X represents: N (64, 2.7) means: (a) Use the 68 − 95 − 99.7 rule to shade the following probability density curves for the normal distribution of heights of women. i. 68% of the data is within one standard deviation of the mean, so the area representing the probability that a randomly selected woman is within one standard deviation of the mean is . ii. 95% of the data is within two standard deviations of the mean, so the area representing the probability that a randomly selected . woman is within two standard deviation of the mean is 1 iii. 99.7% of the data is within three standard deviations of the mean, so the area representing the probability that a randomly selected woman is within three standard deviation of the mean . is Sidetrack: To represent 100% of the data, we would shade in the entire curve, so what is the total area of any probability density curve? 2 (b) What percentage of women are 64 inches or taller? What is the probability that a randomly selected woman will be 64 inches or taller? Be sure to write your result using probability notation with an inequality. (c) What is the probability that a randomly selected woman will be between 61.3 and 66.7 inches tall? As always, be sure to show your work by shading in the area corresponding to the probability in the density curve and then be sure to write your result using probability notation with an inequality. 3 (d) What is the probability that a randomly selected woman will be between 55.9 and 72.1 inches tall? (e) What is the probability that a randomly selected woman will be between 58.6 and 72.1 inches tall? 4 (f) What is the probability that a randomly selected woman will be shorter than 71 inches tall? Note, please stop after attempting this problem. 5 Here are the generic calculator steps for using your graphing calculator to find the shaded area on a density curve that represents a probability for a normally distributed random variable. How to use a TI-83 or TI-84 graphing calculator to find an area on a density curve that represents a probability for a normally distributed random variable: • Press 2nd and then Vars (to get the DISTR menu) • Select normalcdf( (WARNING never select normalpdf) • Complete the entry to obtain normalcdf(left bound, right bound, Mean, Standard Deviation) by substituting in the appropriate values and closing the parentheses. Press ENTER Working together as a class, lets go back to the previous problem and apply our new calculator skills to answer the question. 2. An average light bulb manufactured by the Acme Corporation lasts 300 days with a standard deviation of 50 days. Assuming that bulb life is normally distributed. (a) What is the probability that an Acme light bulb will last at most 365 days? (b) What is the probability that an Acme light bulb will last at least 14 months (assume one month is 30 days)? Be sure to write the probability notation with an inequality. 6 (c) What is the probability that an Acme light bulb will last between 6 and 18 months (assume one month is 30 days)? Now lets reverse the process. So far, weve been given data values and used our calculators to find the probabilities. But now, well be given the probabilities (i.e. the area of the shaded region on a normal density curve) and use our TI-83 or TI-84 graphing calculators to find the data values (i.e. the boundaries of our shaded regions). Here are the steps. (a) Press 2nd and then Vars (to get the DISTR menu) (b) Select invNorm( (c) Complete the entry to obtain invNorm(Area of Shaded Region to the Left, Mean, Standard Deviation) by substituting in the appropriate values and closing the parentheses. (d) Press ENTER Note: invNorm will only give you the boundary (i.e. the data value) for the shaded region to the left of the boundary. In other words invNorm will find the boundary for the shaded region in either of the graph below. −3 −2 | 0 X −1 7 1 2 3 3. Suppose scores on an IQ test are normally distributed with a mean of 100 and a standard deviation of 10. (a) Find the range of scores for the bottom 20% of students? (b) The range of scores for the top 10% of students? (c) The range of scores for the middle 70%? 8 4. In a survey of men in the United States (ages 20 to 29), the mean height was 69.9 inches with a standard deviation of 3.0 inches. (a) What is the probability that a randomly selected man is shorter than 6 feet? (b) How tall are the tallest 20% of men? (c) What is the probability that a randomly selected man will be taller than 5 feet 9 inches? 9 (d) What height represents the 25th percentile? (e) What is the probability that a randomly selected man is between 5 feet 9 inches and 6 feet 6 inches tall? (f) How tall are the middle 75% of men? 10