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To apply for the PGA Associate Program, an applicant must pass a Playing Ability Test. Past data show the total strokes shot by applicants are normally distributed with a mean of 154.2 strokes and a standard deviation of 3.6 strokes. (a) What is the likelihood a random applicant shoots between 148 strokes and 152 strokes? (b) Applicants who shoot in the lowest 2.5% of stroke totals are given preferential status for acceptance into the program. What is the cutoff to qualify for this status?

To apply for the PGA Associate Program, an applicant must pass a Playing Ability Test. Past data show the total strokes shot by applicants are normally distributed with a mean of 154.2 strokes and a standard deviation of 3.6 strokes. (a) What is the likelihood a random applicant shoots between 148 strokes and 152 strokes? (b) Applicants who shoot in the lowest 2.5% of stroke totals are given preferential status for acceptance into the program. What is the cutoff to qualify for this status?

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Question 9.1 - Sampling Distribution for the Mean Please solve the problem with simple probability rules
**Exercise 9. Predicting Performance for PGA Associate Program Admission** To apply for the PGA Associate Program, an applicant must pass a Playing Ability Test. Statistical data from previous applicants indicates that the total strokes shot by applicants follow a normal distribution, with a mean of 154.2 strokes and a standard deviation of 3.6 strokes. ### Problem Breakdown: (a) **Probability Calculation**: - **Question**: What is the likelihood that a random applicant shoots between 148 strokes and 152 strokes? - **Objective**: To determine the probability within a specified range using the properties of the normal distribution. (b) **Cutoff Calculation for Preferential Status**: - **Question**: Applicants who shoot in the lowest 2.5% of stroke totals are given preferential status for acceptance into the program. What is the cutoff to qualify for this preferential status? - **Objective**: To identify the score below which 2.5% of applicant strokes fall, which will require using z-scores and the characteristics of the normal distribution. ### Detailed Explanation and Calculation Steps: 1. **Normal Distribution Parameters**: - Mean (μ) = 154.2 strokes - Standard Deviation (σ) = 3.6 strokes 2. **Calculating Probability (Part a)**: - To find the probability that a score lies between 148 and 152, convert these values to their corresponding z-scores: - \( Z = \frac{X - \mu}{\sigma} \) - Compute the z-scores for X = 148 and X = 152. - Use standard normal distribution tables or software to find the probabilities corresponding to these z-scores. - Subtract the cumulative probability up to 148 strokes from the cumulative probability up to 152 strokes to find the desired probability. 3. **Determining Cutoff for Top 2.5% (Part b)**: - The bottom 2.5% in a normal distribution corresponds to a z-score approximately at -1.96. - Use the z-score formula to convert this into the actual stroke count: - \( X = \mu + (Z \times \sigma) \) - Plug in the mean, standard deviation, and z-score to calculate the cutoff score. ### Hypothetical Example Calculations: 1. **Z-Score for 148 strokes**:
Transcribed Image Text:**Exercise 9. Predicting Performance for PGA Associate Program Admission** To apply for the PGA Associate Program, an applicant must pass a Playing Ability Test. Statistical data from previous applicants indicates that the total strokes shot by applicants follow a normal distribution, with a mean of 154.2 strokes and a standard deviation of 3.6 strokes. ### Problem Breakdown: (a) **Probability Calculation**: - **Question**: What is the likelihood that a random applicant shoots between 148 strokes and 152 strokes? - **Objective**: To determine the probability within a specified range using the properties of the normal distribution. (b) **Cutoff Calculation for Preferential Status**: - **Question**: Applicants who shoot in the lowest 2.5% of stroke totals are given preferential status for acceptance into the program. What is the cutoff to qualify for this preferential status? - **Objective**: To identify the score below which 2.5% of applicant strokes fall, which will require using z-scores and the characteristics of the normal distribution. ### Detailed Explanation and Calculation Steps: 1. **Normal Distribution Parameters**: - Mean (μ) = 154.2 strokes - Standard Deviation (σ) = 3.6 strokes 2. **Calculating Probability (Part a)**: - To find the probability that a score lies between 148 and 152, convert these values to their corresponding z-scores: - \( Z = \frac{X - \mu}{\sigma} \) - Compute the z-scores for X = 148 and X = 152. - Use standard normal distribution tables or software to find the probabilities corresponding to these z-scores. - Subtract the cumulative probability up to 148 strokes from the cumulative probability up to 152 strokes to find the desired probability. 3. **Determining Cutoff for Top 2.5% (Part b)**: - The bottom 2.5% in a normal distribution corresponds to a z-score approximately at -1.96. - Use the z-score formula to convert this into the actual stroke count: - \( X = \mu + (Z \times \sigma) \) - Plug in the mean, standard deviation, and z-score to calculate the cutoff score. ### Hypothetical Example Calculations: 1. **Z-Score for 148 strokes**:
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