A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN: 9780134753119
Author: Sheldon Ross
Publisher: PEARSON
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**Estimating Mean IQ Score and Probability**

The IQ scores for a local high school are approximately normally distributed. Within this curve, 99.7% of the scores, centered about the mean, are between 55 and 145 points. The standard deviation is 15 points. Use this information to estimate the mean score of the IQ test and approximate the probability that a score is less than 70 points. Explain how you determined your answers.

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**Approach:**

1. **Estimate the Mean:**
   - The range 55 to 145 covers 99.7% of the distribution, which represents ±3 standard deviations (SD) from the mean in a normal distribution.
   - Given that each SD is 15 points, calculate the mean:
     - Mean (µ) = (Lower Limit + Upper Limit) / 2 = (55 + 145) / 2 = 100.

2. **Calculate Probability for Scores Less than 70:**
   - Determine the Z-score for 70:
     - Z = (X - µ) / SD = (70 - 100) / 15 = -2.
   - Use a standard normal distribution table to find the probability for Z = -2.
   - The probability of Z being less than -2 is approximately 0.0228 (or 2.28%).

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This information will help in understanding how standard deviation and normal distribution work to analyze statistical data in educational settings.
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Transcribed Image Text:**Estimating Mean IQ Score and Probability** The IQ scores for a local high school are approximately normally distributed. Within this curve, 99.7% of the scores, centered about the mean, are between 55 and 145 points. The standard deviation is 15 points. Use this information to estimate the mean score of the IQ test and approximate the probability that a score is less than 70 points. Explain how you determined your answers. --- **Approach:** 1. **Estimate the Mean:** - The range 55 to 145 covers 99.7% of the distribution, which represents ±3 standard deviations (SD) from the mean in a normal distribution. - Given that each SD is 15 points, calculate the mean: - Mean (µ) = (Lower Limit + Upper Limit) / 2 = (55 + 145) / 2 = 100. 2. **Calculate Probability for Scores Less than 70:** - Determine the Z-score for 70: - Z = (X - µ) / SD = (70 - 100) / 15 = -2. - Use a standard normal distribution table to find the probability for Z = -2. - The probability of Z being less than -2 is approximately 0.0228 (or 2.28%). --- This information will help in understanding how standard deviation and normal distribution work to analyze statistical data in educational settings.
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