\( 1 \leftarrow \) Scores of an IQ test have a bell-shaped distribution with a mean of 100 and a standard deviation of 17 . Use the empirical rule to determine the following (a) What percentage of people has an IQ score between 66 and 134 ? (b) What percentage of people has an IQ score less than 83 or greater than 117? (c) What percentage of people has an IQ score greater than 151 ?

In a bell-shaped distribution, also known as a normal distribution, the empirical rule states that approximately 68% of the data falls within one standard deviation of the mean, about 95% within two standard deviations, and around 99.7% within three standard deviations. In this case, with a mean of 100 and a standard deviation of 17: (a) For an IQ score between 66 and 134, we calculate one standard deviation below the mean (100 - 17 = 83) and one standard deviation above the mean (100 + 17 = 117). Thus, the range of 66 (100 - 2*17) to 134 (100 + 2*17) covers two standard deviations. By the empirical rule, about 95% of people fall within this range. (b) The scores less than 83 (mean - 1 sd) and greater than 117 (mean + 1 sd) are considered "outside" the range of 68% in the middle. Since 68% of people score between 83 and 117, that leaves 32% outside of this range. However, since the normal distribution is symmetric, this means there are 16% below 83 and 16% above 117, for a total of 32%. (c) For scores greater than 151, we determine how many standard deviations this score is from the mean. The score of 151 is 3 standard deviations above the mean (100 + 3*17). According to the empirical rule, only about 0.15% of the population scores greater than 151, which means it's quite a rare occurrence!