MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN: 9781119256830
Author: Amos Gilat
Publisher: John Wiley & Sons Inc
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### Finding Probability in a Normally Distributed Population Assume a member is selected at random from the population represented by the graph. Find the probability that the member selected at random is from the shaded area of the graph. Assume the variable \(x\) is normally distributed. **Graph Description:** The graph shows the normal distribution of SAT Critical Reading Scores, which are normally distributed with a mean (\(\mu\)) of 509 and a standard deviation (\(\sigma\)) of 119. The shaded region represents the scores between 200 and 375. - **X-axis:** Represents the SAT Critical Reading Scores, ranging from 200 to 800. - **Y-axis (not labeled but implied):** Represents the probability density. The probabilities are visually indicated by the area under the curve between the specified scores. The distribution curve indicates the population's normal distribution, with most values clustering around the mean score of 509. **Calculation Task:** The task is to find the probability that a randomly selected member’s SAT Critical Reading Score falls within the shaded region (200 < \(x\) < 375). **Solution:** Let’s denote the desired probability as \( P \): \[ P = P(200 < x < 375) \] This probability can be found using the cumulative distribution function (CDF) of the normal distribution: 1. **Calculate the Z-scores:** \[ Z_1 = \frac{200 - \mu}{\sigma} = \frac{200 - 509}{119} \approx -2.6 \] \[ Z_2 = \frac{375 - \mu}{\sigma} = \frac{375 - 509}{119} \approx -1.13 \] 2. **Look up the cumulative probabilities for these Z-scores.** \[ P(Z_1 \approx -2.6) \approx 0.0047 \] \[ P(Z_2 \approx -1.13) \approx 0.1292 \] 3. **Find the probability for the desired range by subtracting these cumulative probabilities:** \[ P(200 < x < 375) = P(Z_2) - P(Z_1) \approx 0.1292 - 0.0047 \approx 0.1245 \] \[ \boxed{0.1245} \] The probability that the member selected at random is from the
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Transcribed Image Text:### Finding Probability in a Normally Distributed Population Assume a member is selected at random from the population represented by the graph. Find the probability that the member selected at random is from the shaded area of the graph. Assume the variable \(x\) is normally distributed. **Graph Description:** The graph shows the normal distribution of SAT Critical Reading Scores, which are normally distributed with a mean (\(\mu\)) of 509 and a standard deviation (\(\sigma\)) of 119. The shaded region represents the scores between 200 and 375. - **X-axis:** Represents the SAT Critical Reading Scores, ranging from 200 to 800. - **Y-axis (not labeled but implied):** Represents the probability density. The probabilities are visually indicated by the area under the curve between the specified scores. The distribution curve indicates the population's normal distribution, with most values clustering around the mean score of 509. **Calculation Task:** The task is to find the probability that a randomly selected member’s SAT Critical Reading Score falls within the shaded region (200 < \(x\) < 375). **Solution:** Let’s denote the desired probability as \( P \): \[ P = P(200 < x < 375) \] This probability can be found using the cumulative distribution function (CDF) of the normal distribution: 1. **Calculate the Z-scores:** \[ Z_1 = \frac{200 - \mu}{\sigma} = \frac{200 - 509}{119} \approx -2.6 \] \[ Z_2 = \frac{375 - \mu}{\sigma} = \frac{375 - 509}{119} \approx -1.13 \] 2. **Look up the cumulative probabilities for these Z-scores.** \[ P(Z_1 \approx -2.6) \approx 0.0047 \] \[ P(Z_2 \approx -1.13) \approx 0.1292 \] 3. **Find the probability for the desired range by subtracting these cumulative probabilities:** \[ P(200 < x < 375) = P(Z_2) - P(Z_1) \approx 0.1292 - 0.0047 \approx 0.1245 \] \[ \boxed{0.1245} \] The probability that the member selected at random is from the
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