Question 1200367: At a large publishing company, the mean age of proofreaders is 36.2 years and the standard deviation of the distribution is 3.7 years. A random sample of 40 proofreaders is selected.
a. State the importance of central limit theorem if we are interested to determine the probability for the sample mean age of the selected proofreaders. (2 marks)
b. Find the probability that the mean age of the proofreaders in the sample will be between 35 and 37 years.
Answer by textot(100)   (Show Source):
You can put this solution on YOUR website! **a. Importance of the Central Limit Theorem**
The Central Limit Theorem (CLT) is crucial when determining the probability for the sample mean age of the selected proofreaders. Here's why:
* **Approximation to Normal Distribution:** Even if the population distribution of proofreaders' ages is not perfectly normal, the CLT states that the distribution of sample means will tend towards a normal distribution as the sample size increases. In this case, with a sample size of 40, we can reasonably assume that the distribution of sample means will be approximately normal.
* **Calculation of Probabilities:** The CLT allows us to use the properties of the normal distribution (mean and standard deviation) to calculate probabilities related to the sample mean. This is essential for answering questions like "What is the probability that the mean age of the sample will fall within a specific range?"
**b. Probability of Sample Mean Age Between 35 and 37 Years**
1. **Calculate Standard Error of the Mean:**
- Standard Error (SE) = σ / √n
- SE = 3.7 / √40
- SE ≈ 0.585
2. **Calculate Z-scores:**
- Z1 = (35 - 36.2) / 0.585 ≈ -2.05
- Z2 = (37 - 36.2) / 0.585 ≈ 1.37
3. **Find Probabilities Using Standard Normal Distribution:**
- Use a standard normal distribution table or a calculator to find the probabilities associated with these Z-scores:
- P(Z < -2.05) ≈ 0.0202
- P(Z < 1.37) ≈ 0.9147
4. **Calculate the Probability Between 35 and 37:**
- P(35 < X̄ < 37) = P(Z < 1.37) - P(Z < -2.05)
- P(35 < X̄ < 37) ≈ 0.9147 - 0.0202
- P(35 < X̄ < 37) ≈ 0.8945
**Therefore, the probability that the mean age of the proofreaders in the sample will be between 35 and 37 years is approximately 0.8945.**
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