Section 7.1 The Standard Normal Curve 297 EXCEL Finding areas under the standard normal curve The following procedure computes the area to the left of a given value. Step 1. In an empty cell, select the Insert Function icon and highlight Statistical in the category field. Step 2. Click on the NORM.S.DIST function and press OK. Step 3. To compute the area to the left of a given x, enter the value of x in the X field. Step 4. Enter TRUE in the Cumulative field and click OK. Figure J illustrates finding the area to the left of z = 1.26 (Example 7.2). Figure J Finding a z-score corresponding to a given area The following procedure is used to calculate the z-score corresponding to an area to the left. Step 1. In an empty cell, select the Insert Function icon and highlight Statistical in the category field. Step 2. Click on the NORM.S.INV function and press OK. Step 3. Enter the area to the left of the desired z-score in the Probability field. Step 4. Click OK. Figure K illustrates finding the z-score that has an area of 0.26 to its left (Example 7.8). Figure K SECTION 7.1 Exercises Exercises 1– 8 are the Check Your Understanding exercises located within the section. Understanding the Concepts In Exercises 9–14, fill in each blank with the appropriate word or phrase. 9. The proportion of a population that is contained within an interval corresponds to an area under the probability curve. density 10. If X is a continuous random variable, then P(X = a) = for any number a. 0 11. The area under the entire probability density curve is equal to . 1 12. The mean, median, and mode of a normal distribution are to each other. equal 13. A normal distribution with mean 0 and standard deviation 1 is called the normal distribution. standard 14. Points on the horizontal axis to the left of the mode have z-scores. negative In Exercises 15–18, determine whether the statement is true or false. If the statement is false, rewrite it as a true statement. 15. The probability that a randomly selected value of a continuous random variable lies between a and b is given by the area under the probability density curve between a and b. True 16. A normal curve is symmetric around its mode. True 17. A normal curve is wide and flat when the standard deviation is small. False 18. The area under the normal curve to the left of the mode is less than 0.5. False Practicing the Skills 19. The following figure is a probability density curve that represents the lifetime, in months, of a certain type of laptop battery. Area = 0.35 Area = 0.25 0 6 12 18 24 30 36 42 48 a. Find the proportion of batteries with lifetimes between 12 and 18 months. 0.25 b. Find the proportion of batteries with lifetimes less than 18 months. 0.6 c. What is the probability that a randomly chosen battery lasts more than 18 months? 0.4 20. The following figure is a probability density curve that represents the grade point averages (GPA) of the graduating seniors at a large university.
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