# Probability Calculations for Customer Satisfaction Given: - Proportion of satisfied customers, \( p = 0.70 \) - Sample size, \( n = 74 \) Let \( X \) be the number of satisfied customers in the sample. \( X \) follows a binomial distribution: \[ X \sim \text{Binomial}(n=74, p=0.70) \] For large \( n \), we can use the normal approximation to the binomial. --- ## Step 1: Find Mean and Standard Deviation - **Mean**: \[ \mu = np = 74 \times 0.70 = 51.8 \] - **Standard deviation**: \[ \sigma = \sqrt{np(1-p)} = \sqrt{74 \times 0.70 \times 0.30} = \sqrt{15.54} \approx 3.9447 \] --- ## Step 2: Normal Approximation with Continuity Correction ### (a) Probability that **less than 52** customers are satisfied (\( P(X < 52) \)) - Use continuity correction: \( X < 52 \) becomes \( X \leq 51.5 \) - Calculate z-score: \[ z = \frac{51.5 - 51.8}{3.9447} = \frac{-0.3}{3.9447} \approx -0.08 \] - Find the probability using the standard normal table: \[ P(X < 52) \approx P(Z < -0.08) = 0.4681 \] --- ### (b) Probability that **at least 52** customers are satisfied (\( P(X \geq 52) \)) - Use continuity correction: \( X \geq 52 \) becomes \( X \geq 51.5 \) - Calculate z-score: \[ z = \frac{51.5 - 51.8}{3.9447} \approx -0.08 \] - Probability: \[ P(X \geq 52) = 1 - P(X < 51.5) = 1 - 0.4681 = 0.5319 \] --- ### (c) Probability that the sample proportion is **between 68% and 74%** (\( 0.68 < \hat{p} < 0.74 \)) - The sample proportion \( \hat{p} = \frac{X}{n} \) - \( 0.68 < \frac{X}{74} < 0.74 \) - Convert to \( X \): \( 0.68 \times 74 = 50.32 \), \( 0.74 \times 74 = 54.76 \) - Apply continuity correction: - Lower bound: \( X \geq 50.5 \) - Upper bound: \( X \leq 54.5 \) #### Find z-scores: - For \( X = 50.5 \): \[ z_1 = \frac{50.5 - 51.8}{3.9447} = \frac{-1.3}{3.9447} \approx -0.33 \] - For \( X = 54.5 \): \[ z_2 = \frac{54.5 - 51.8}{3.9447} = \frac{2.7}{3.9447} \approx 0.68 \] #### Find probabilities: - \( P(z_1 < Z < z_2) = P(Z < 0.68) - P(Z < -0.33) \) - From the standard normal table: - \( P(Z < 0.68) = 0.7517 \) - \( P(Z < -0.33) = 0.3707 \) \[ P(0.68 < \hat{p} < 0.74) = 0.7517 - 0.3707 = 0.3810 \] --- ## **Summary Table** | Part | Calculation | Z-value(s) | Probability | |------|-------------|------------|-------------| | (a) | \( P(X < 52) \) | -0.08 | **0.4681** | | (b) | \( P(X \geq 52) \) | -0.08 | **0.5319** | | (c) | \( P(0.68 < \hat{p} < 0.74) \) | -0.33, 0.68 | **0.3810** | --- ## **Conclusion** - (a) **Probability less than 52 satisfied:** 0.4681 - (b) **Probability at least 52 satisfied:** 0.5319 - (c) **Probability proportion is between 68% and 74%:** 0.3810 --- **Note:** Values are rounded as requested. **No image is included as this is a calculation-based problem.**