SOLUTION: Based on historical data, your manager believes that 38% of the company's orders come from first-time customers. A random sample of 229 orders will be used to estimate the proporti

Click here to see ALL problems on Probability-and-statistics

Question 1186293: Based on historical data, your manager believes that 38% of the company's orders come from first-time customers. A random sample of 229 orders will be used to estimate the proportion of first-time-customers. What is the probability that the sample proportion is between 0.3 and 0.4?
Note: You should carefully round any z-values you calculate to 4 decimal places to match wamap's approach and calculations.
Answer by CPhill(1959) (Show Source):
You can put this solution on YOUR website!
Here's how to solve this problem using the normal approximation to the binomial distribution:
1. **Check if the normal approximation is appropriate:**
* n * p = 229 * 0.38 = 86.82 >= 10
* n * (1-p) = 229 * (1 - 0.38) = 229 * 0.62 = 142.18 >= 10
Since both conditions are met, the normal approximation is reasonable.
2. **Calculate the mean (μ) and standard deviation (σ) of the sample proportion:**
* μ = p = 0.38
* σ = sqrt[p * (1-p) / n] = sqrt[0.38 * 0.62 / 229] ≈ 0.0322
3. **Calculate the z-scores:**
The z-score formula for proportions is: z = (p̂ - p) / σ
* For p̂ = 0.3: z = (0.3 - 0.38) / 0.0322 ≈ -2.4845
* For p̂ = 0.4: z = (0.4 - 0.38) / 0.0322 ≈ 0.6211
4. **Find the probabilities using the z-table or calculator:**
* P(z < -2.4845) ≈ 0.0065 (This is the area to the left of -2.4845)
* P(z < 0.6211) ≈ 0.7327 (This is the area to the left of 0.6211)
5. **Calculate the probability between 0.3 and 0.4:**
P(0.3 < p̂ < 0.4) = P(z < 0.6211) - P(z < -2.4845)
P(0.3 < p̂ < 0.4) ≈ 0.7327 - 0.0065
P(0.3 < p̂ < 0.4) ≈ 0.7262
Therefore, the probability that the sample proportion is between 0.3 and 0.4 is approximately **0.7262**.