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 strokes2. **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**:

From the provided information,Mean (µ) = 154.2Standard deviation (σ) = 3.6X~N (154.2, 3.6)

Answered: To apply for the PGA Associate Program,…

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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**:

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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