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A school gives an entry exam for admission. Suppose the score of this exam follows a normal distribution N(400, 60). This year, the school decides to admit students who score in the top 30%. Suppose a student scored 428 on the test. Will the student be admitted? Explain your reasoning.

A school gives an entry exam for admission. Suppose the score of this exam follows a normal distribution N(400, 60). This year, the school decides to admit students who score in the top 30%. Suppose a student scored 428 on the test. Will the student be admitted? Explain your reasoning.

Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter10: Statistics
Section10.4: Distributions Of Data
Problem 22PFA
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**Admission Criteria Based on Entry Exam Scores** A school gives an entry exam for admission. Suppose the score of this exam follows a normal distribution N(400, 60). This year, the school decides to admit students who score in the top 30%. Suppose a student scored 428 on the test. Will the student be admitted? Explain your reasoning. --- In this context, we are dealing with a normal distribution with a mean (μ) of 400 and a standard deviation (σ) of 60. We need to find out if a score of 428 is within the top 30% of this distribution. ### Steps to Determine Admission: 1. **Convert the Score to a Z-Score:** The Z-score formula is given by: \[ Z = \frac{(X - \mu)}{\sigma} \] Where: - \( X \) is the raw score (428 in this case). - \( \mu \) is the mean score (400). - \( \sigma \) is the standard deviation (60). Substituting the values: \[ Z = \frac{(428 - 400)}{60} \approx 0.467 \] 2. **Find the Percentile:** The Z-score of 0.467 corresponds to a percentile in the standard normal distribution table. Using a Z-table or statistical software, a Z-score of 0.467 corresponds approximately to the 68th percentile. 3. **Determine Admission:** The school admits students who score in the top 30%. This means that these students need to be in the 70th percentile or above (since 100% - 30% = 70%). Since a score of 428 corresponds to the 68th percentile, which is below the 70th percentile threshold, the student will not be admitted. ### Final Answer: No, the student will not be admitted, as their score of 428 does not fall within the top 30% of the distribution.
Transcribed Image Text:**Admission Criteria Based on Entry Exam Scores** A school gives an entry exam for admission. Suppose the score of this exam follows a normal distribution N(400, 60). This year, the school decides to admit students who score in the top 30%. Suppose a student scored 428 on the test. Will the student be admitted? Explain your reasoning. --- In this context, we are dealing with a normal distribution with a mean (μ) of 400 and a standard deviation (σ) of 60. We need to find out if a score of 428 is within the top 30% of this distribution. ### Steps to Determine Admission: 1. **Convert the Score to a Z-Score:** The Z-score formula is given by: \[ Z = \frac{(X - \mu)}{\sigma} \] Where: - \( X \) is the raw score (428 in this case). - \( \mu \) is the mean score (400). - \( \sigma \) is the standard deviation (60). Substituting the values: \[ Z = \frac{(428 - 400)}{60} \approx 0.467 \] 2. **Find the Percentile:** The Z-score of 0.467 corresponds to a percentile in the standard normal distribution table. Using a Z-table or statistical software, a Z-score of 0.467 corresponds approximately to the 68th percentile. 3. **Determine Admission:** The school admits students who score in the top 30%. This means that these students need to be in the 70th percentile or above (since 100% - 30% = 70%). Since a score of 428 corresponds to the 68th percentile, which is below the 70th percentile threshold, the student will not be admitted. ### Final Answer: No, the student will not be admitted, as their score of 428 does not fall within the top 30% of the distribution.
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