Section 7.2 Applications of the Normal Distribution 301 Check Your Understanding 1. A normal distribution has mean �� = 15 and standard deviation �� = 4. Find and interpret the z-score for x = 11. z = −1 2. A normal distribution has mean �� = 60 and standard deviation �� = 20. Find and interpret the z-score for x = 75. z = 0.75 3. Compact fluorescent bulbs are more energy efficient than incandescent bulbs, but they take longer to reach full brightness. The time that it takes for a compact fluorescent bulb to reach full brightness is normally distributed with mean 29.8 seconds and standard deviation 4.5 seconds. A randomly selected bulb takes 28 seconds to reach full brightness. Find and interpret the z-score for x = 28. z = −0.4 Answers are on page 310. Objective 2 Find areas under a normal curve Finding Areas Under a Normal Curve In Section 7.1, we used z-scores to compute areas under the standard normal curve. By standardizing, we can use z-scores to compute areas under a normal curve with any mean and standard deviation. An area under a normal curve over an interval can be interpreted in two ways: It represents the proportion of the population that is contained within the interval, and it also represents the probability that a randomly selected individual will have a value within the interval. EXAMPLE 7.13 Finding an area under a normal curve Explain It Again Probabilities and proportions: The probability that a randomly sampled value falls in a given interval is equal to the proportion of the population that is contained in the interval. So the area under a normal curve represents both probabilities and proportions. A study reported that the length of pregnancy from conception to birth is approximately normally distributed with mean �� = 272 days and standard deviation �� = 9 days. What proportion of pregnancies last longer than 280 days? Source: Singapore Medical Journal 35:1044–1048 Solution The proportion of pregnancies lasting longer than 280 days is equal to the area under the normal curve corresponding to values of x greater than 280. We find this area as follows. Step 1: Find the z-score for x = 280. z = x − �� �� = 280 − 272 9 = 0.89 Step 2: Sketch a normal curve, label the mean, x-value, and z-score, and shade in the area to be found. See Figure 7.15. 272 280 z = 0.89 Figure 7.15 Step 3: Find the area to the right of z = 0.89. Using Table A.2, we find the area to the left of z = 0.89 to be 0.8133. The area to the right is therefore 1 − 0.8133 = 0.1867. We conclude that the proportion of pregnancies that last longer than 280 days is 0.1867.
navidi_monk_elementary_statistics_2e_ch7-9
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