SECTION 5.1 Introduction to Normal Distributions and the Standard Normal Distribution 245 Finding Probability In Exercises 47–56, find the indicated probability using the standard normal distribution. If convenient, use technology to find the probability. 47. P1z 6 0.532 48. P1z 6 -1.112 49. P1z 7 2.1752 50. P1z 7 -1.852 51. P1-0.89 6 z 6 02 52. P10 6 z 6 0.8352 53. P1-1.78 6 z 6 1.782 54. P1-1.54 6 z 6 1.542 55. P1z 6 -2.58 or z 7 2.582 56. P1z 6 -1.22 or z 7 1.322 Extending Concepts 57. Writing Draw a normal curve with a mean of 60 and a standard deviation of 12. Describe how you constructed the curve and discuss its features. 58. Writing Draw a normal curve with a mean of 450 and a standard deviation of 50. Describe how you constructed the curve and discuss its features. Uniform Distribution A uniform distribution is a continuous probability distribution for a random variable x between two values a and b 1a 6 b2, where a … x … b and all of the values of x are equally likely to occur. The graph of a uniform distribution is shown below. a b 1 b − a x y The probability density function of a uniform distribution is y = 1 b - a on the interval from x = a to x = b. For any value of x less than a or greater than b, y = 0. In Exercises 59 and 60, use this information. 59. Show that the probability density function of a uniform distribution satisfies the two conditions for a probability density function. 60. For two values c and d, where a … c 6 d … b, the probability that x lies between c and d is equal to the area under the curve between c and d, as shown below. a c d b 1 b − a x y So, the area of the red region equals the probability that x lies between c and d. For a uniform distribution from a = 1 to b = 25, find the probability that (a) x lies between 2 and 8. (b) x lies between 4 and 12. (c) x lies between 5 and 17. (d) x lies between 8 and 14.
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