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The IQs of 500 applicants to a certain college are approximately normally distributed with a mean of 118 and a standard deviation of 11. If the college requires an IQ of at least 98, how many of these students will be rejected on this basis of IQ, regardless of their other qualifications? Note that IQs are recorded to the nearest integers. Click here to view page 1 of the standard normal distribution table. Click here to view page 2 of the standard normal distribution table. student(s) (Round to the nearest whole number as needed.)

The IQs of 500 applicants to a certain college are approximately normally distributed with a mean of 118 and a standard deviation of 11. If the college requires an IQ of at least 98, how many of these students will be rejected on this basis of IQ, regardless of their other qualifications? Note that IQs are recorded to the nearest integers. Click here to view page 1 of the standard normal distribution table. Click here to view page 2 of the standard normal distribution table. student(s) (Round to the nearest whole number as needed.)

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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**Problem Statement:** The IQs of 500 applicants to a certain college are approximately normally distributed with a mean of 118 and a standard deviation of 11. If the college requires an IQ of at least 98, how many of these students will be rejected on this basis of IQ, regardless of their other qualifications? Note that IQs are recorded to the nearest integers. [Click here to view page 1 of the standard normal distribution table.](#) [Click here to view page 2 of the standard normal distribution table.](#) --- **Answer Box:** [ ] student(s) (Round to the nearest whole number as needed.) --- **Explanation:** To solve this problem, we need to determine the proportion of IQ scores that fall below 98 in a normal distribution with a mean of 118 and a standard deviation of 11. Using the standard normal distribution table (often referred to as the Z-table), we need to convert the IQ score to a Z-score and then find the corresponding probability. **Step-by-Step Solution:** 1. **Calculate the Z-score** for an IQ of 98: \[ Z = \frac{X - \mu}{\sigma} \] where \( X \) is the IQ score, \( \mu \) is the mean, and \( \sigma \) is the standard deviation. \[ Z = \frac{98 - 118}{11} \approx -1.82 \] 2. **Look up the Z-score** in the standard normal distribution table to find the probability: The Z-score of -1.82 corresponds to a cumulative probability of approximately 0.0344. This means that approximately 3.44% of applicants have an IQ less than 98. 3. **Calculate the number of applicants**: \[ \text{Number of rejected students} = 0.0344 \times 500 \approx 17.2 \] Rounding to the nearest whole number, approximately 17 applicants will be rejected based on having an IQ less than 98. **Answer:** 17 students (Round to the nearest whole number as needed.)
Transcribed Image Text:**Problem Statement:** The IQs of 500 applicants to a certain college are approximately normally distributed with a mean of 118 and a standard deviation of 11. If the college requires an IQ of at least 98, how many of these students will be rejected on this basis of IQ, regardless of their other qualifications? Note that IQs are recorded to the nearest integers. [Click here to view page 1 of the standard normal distribution table.](#) [Click here to view page 2 of the standard normal distribution table.](#) --- **Answer Box:** [ ] student(s) (Round to the nearest whole number as needed.) --- **Explanation:** To solve this problem, we need to determine the proportion of IQ scores that fall below 98 in a normal distribution with a mean of 118 and a standard deviation of 11. Using the standard normal distribution table (often referred to as the Z-table), we need to convert the IQ score to a Z-score and then find the corresponding probability. **Step-by-Step Solution:** 1. **Calculate the Z-score** for an IQ of 98: \[ Z = \frac{X - \mu}{\sigma} \] where \( X \) is the IQ score, \( \mu \) is the mean, and \( \sigma \) is the standard deviation. \[ Z = \frac{98 - 118}{11} \approx -1.82 \] 2. **Look up the Z-score** in the standard normal distribution table to find the probability: The Z-score of -1.82 corresponds to a cumulative probability of approximately 0.0344. This means that approximately 3.44% of applicants have an IQ less than 98. 3. **Calculate the number of applicants**: \[ \text{Number of rejected students} = 0.0344 \times 500 \approx 17.2 \] Rounding to the nearest whole number, approximately 17 applicants will be rejected based on having an IQ less than 98. **Answer:** 17 students (Round to the nearest whole number as needed.)
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