2=(2>60) (2<80) 5. A The ages of the thousands of residents of a retirement community are normally distributed with a mean of 70 and a standard deviation of 4 years. a. What proportion of this population is between 60 and 80? Z 2=60-70 = -2.5 = 6.21 4 2-80-70 시 2= = 2.5=6.21 71.5-45 4 99274 1-,99379 -3 b. If one sample of 45 residents is chosen at random, what is the probability that the sample mean age will be between 68.5 and 71.75? 68.5-45 2= 5.87 ماما - = 6.21% +6.21= -4 12.24% is between con 600-80 years old 2.5 C. Between which two symmetric limits are 95% of all the possible values of the sample means? a)
2=(2>60) (2<80) 5. A The ages of the thousands of residents of a retirement community are normally distributed with a mean of 70 and a standard deviation of 4 years. a. What proportion of this population is between 60 and 80? Z 2=60-70 = -2.5 = 6.21 4 2-80-70 시 2= = 2.5=6.21 71.5-45 4 99274 1-,99379 -3 b. If one sample of 45 residents is chosen at random, what is the probability that the sample mean age will be between 68.5 and 71.75? 68.5-45 2= 5.87 ماما - = 6.21% +6.21= -4 12.24% is between con 600-80 years old 2.5 C. Between which two symmetric limits are 95% of all the possible values of the sample means? a)
MATLAB: An Introduction with Applications
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Author:Amos Gilat
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Chapter1: Starting With Matlab
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Transcribed Image Text:### Statistical Analysis of a Retirement Community
#### Problem 5: Age Distribution Analysis
**Ages of Residents:**
The ages of the thousands of residents at a retirement community are normally distributed with:
- **Mean (μ):** 70
- **Standard Deviation (σ):** 4 years
**Objective:** Analyze age distribution and probabilities associated with specific age ranges.
---
**a. Proportion of Population between Ages 60 and 80**
1. **Calculate Z-scores:**
- Age 60: \( Z = \frac{60 - 70}{4} = -2.5 \)
- Age 80: \( Z = \frac{80 - 70}{4} = 2.5 \)
2. **Probability Calculation:**
- The proportion of the population between \( Z \) scores \(-2.5\) and \(2.5\) corresponds to approximately 6.21% + 6.21% = 12.42%.
- Note: The detailed steps include calculating areas under the normal distribution curve for the given Z-scores.
---
**b. Probability for Sample Mean**
- **Sample Size (n):** 45 residents
- Determine probability of sample mean between 68.5 and 71.75 years:
1. **Calculate Z-scores:**
- For 68.5: \( Z = \frac{68.5 - 70}{4 / \sqrt{45}} \approx -3 \)
- For 71.75: \( Z = \frac{71.75 - 70}{4 / \sqrt{45}} \approx +4 \)
2. **Interpretation:**
- Using Z-tables, identify the probability associated with the Z-scores to determine the likelihood of the sample mean falling within this range.
---
**c. Symmetric Limits for 95% Sample Means**
- **Symmetric Limits Identification:**
- Calculate the Z-scores that correspond to the middle 95% of the normal distribution: \(-1.96\) and \(1.96\).
- Convert these Z-scores back to age values to define limits for the sample means.
- **Z-values for population mean:**
- Calculation yields a range displayed as -8 in the handwritten note.
---
**Notes:**
- **Annotations:** Conversions, probabilities, and critical Z-values
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Let X denote the age of a resident of the retirement community. Given that .
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