Section 7.2 Applications of the Normal Distribution 299 37. Find the z-score for which the area to its right is 0.35. 0.39 38. Find the z-score for which the area to its right is 0.92. −1.41 39. Find the z-scores that bound the middle 50% of the area under the standard normal curve. −0.67, 0.67 40. Find the z-scores that bound the middle 70% of the area under the standard normal curve. −1.04, 1.04 41. Find the z-scores that bound the middle 80% of the area under the standard normal curve. −1.28, 1.28 42. Find the z-scores that bound the middle 98% of the area under the standard normal curve. −2.33, 2.33 Working with the Concepts 43. Symmetry: The area under the standard normal curve to the left of z = −1.75 is 0.0401. What is the area to the right of z = 1.75? 0.0401 44. Symmetry: The area under the standard normal curve to the right of z = −0.51 is 0.6950. What is the area to the left of z = 0.51? 0.6950 45. Symmetry: The area under the standard normal curve between z = −1.93 and z = 0.59 is 0.6956. What is the area between z = −0.59 and z = 1.93? 0.6956 46. Symmetry: The area under the standard normal curve between z = 1.32 and z = 1.82 is 0.0590. What is the area between z = −1.82 and z = −1.32? 0.0590 Extending the Concepts 47. No table, no technology: Let a be the number such that the area to the right of z = a is 0.3. Without using a table or technology, find the area to the left of z = −a. 0.3 48. No table, no technology: Let a be the number such that the area to the right of z = a is 0.21. Without using a table or technology, find the area between z = −a and z = a. 0.58 Answers to Check Your Understanding Exercises for Section 7.1 1. a. 0.63 b. 0.23 c. 0.86 d. 0.14 2. 0.5987 3. 0.0104 4. 0.8491 5. −0.13 6. 0.33 7. 0.64 8. 1.34 SECTION 7.2 Applications of the Normal Distribution Objectives 1. Convert values from a normal distribution to z-scores 2. Find areas under a normal curve 3. Find the value from a normal distribution corresponding to a given proportion In Section 7.1, we found areas under a standard normal curve, which has mean 0 and standard deviation 1. In this section, we will learn to find areas under normal curves with any mean and standard deviation. Objective 1 Convert values from a normal distribution to z -scores Converting Normal Values to z-Scores Let x be a value from a normal distribution with mean �� and standard deviation ��. We can convert x to a z-score by using a method known as standardization. To standardize a value, subtract the mean and divide by the standard deviation. This produces the z-score. Recall: We first described the method for finding the z-score in Section 3.3. DEFINITION Let x be a value from a normal distribution with mean �� and standard deviation ��. The z-score of x is z = x − �� �� The z-score satisfies the following properties.
navidi_monk_elementary_statistics_2e_ch7-9
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