Scores of an IQ test have a bell-shaped distribution with a mean of 100 and a standard deviation of 19 . Use the empirical rule to determine the following. (a) What percentage of people has an IQ score between 81 and 119 ? (b) What percentage of people has an IQ score less than 43 or greater than 157 ? (c) What percentage of people has an IQ score greater than 119 ? (a) \( 68 \% \) (Type an integer or a decimal.) (b) \( \square \% \) (Type an integer or a decimal.)

To solve the problem using the empirical rule, we need to understand the distribution of IQ scores, which is bell-shaped (normal distribution) with a mean (\( \mu \)) of 100 and a standard deviation (\( \sigma \)) of 19. ### Empirical Rule Overview The empirical rule states that for a normal distribution: - Approximately \( 68\% \) of the data falls within one standard deviation of the mean (\( \mu \pm \sigma \)). - Approximately \( 95\% \) of the data falls within two standard deviations of the mean (\( \mu \pm 2\sigma \)). - Approximately \( 99.7\% \) of the data falls within three standard deviations of the mean (\( \mu \pm 3\sigma \)). ### Given Information - Mean (\( \mu \)) = 100 - Standard Deviation (\( \sigma \)) = 19 ### (a) Percentage of people with IQ scores between 81 and 119 To find the percentage of people with IQ scores between 81 and 119, we calculate: - Lower bound: \( 81 = 100 - 19 \) (which is \( \mu - \sigma \)) - Upper bound: \( 119 = 100 + 19 \) (which is \( \mu + \sigma \)) According to the empirical rule, approximately \( 68\% \) of the data falls within one standard deviation of the mean. **Answer for (a):** \( 68\% \) ### (b) Percentage of people with IQ scores less than 43 or greater than 157 First, we calculate the z-scores for these values: - For \( 43 \): \[ z = \frac{43 - 100}{19} \approx -3 \] - For \( 157 \): \[ z = \frac{157 - 100}{19} \approx 3 \] According to the empirical rule, approximately \( 99.7\% \) of the data falls within three standard deviations of the mean. Therefore, the percentage of people outside this range (less than 43 or greater than 157) is: \[ 100\% - 99.7\% = 0.3\% \] **Answer for (b):** \( 0.3\% \) ### (c) Percentage of people with IQ scores greater than 119 Since \( 119 \) is one standard deviation above the mean, we know that: - \( 68\% \) of the data is between \( 81 \) and \( 119 \). - Therefore, \( 32\% \) of the data is outside this range (i.e., \( 16\% \) below \( 81 \) and \( 16\% \) above \( 119 \)). Thus, the percentage of people with IQ scores greater than \( 119 \) is \( 16\% \). **Answer for (c):** \( 16\% \) ### Summary of Answers - (a) \( 68\% \) - (b) \( 0.3\% \) - (c) \( 16\% \)