Question 1198350: 3. A market research consultant hired by Coke Classic Company is interested in estimating the difference between the proportions of female and male customers who favor Coke Classic over Pepsi Cola in Chicago. A random sample of 200 consumers from the market under investigation showed the following frequency distribution.
Male Female
Coke 72 38 110
Pepsi 58 32 90
Construct a 95% confidence interval for the difference between the proportions of
male and female customers who prefer Coke Classic over Pepsi Cola.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! **1. Calculate Sample Proportions**
* **Proportion of Male Customers Preferring Coke:**
* p1 = (Number of Male Customers Preferring Coke) / (Total Number of Male Customers)
* p1 = 72 / 130
* p1 = 0.5538
* **Proportion of Female Customers Preferring Coke:**
* p2 = (Number of Female Customers Preferring Coke) / (Total Number of Female Customers)
* p2 = 38 / 70
* p2 = 0.5429
**2. Calculate the Difference in Sample Proportions**
* p̂1 - p̂2 = 0.5538 - 0.5429
* p̂1 - p̂2 = 0.0109
**3. Calculate Pooled Proportion**
* Pooled Proportion (p̂) = [(Number of Male Customers Preferring Coke) + (Number of Female Customers Preferring Coke)] / (Total Number of Male Customers + Total Number of Female Customers)
* p̂ = (72 + 38) / (130 + 70)
* p̂ = 110 / 200
* p̂ = 0.55
**4. Calculate Standard Error**
* Standard Error (SE) = √[p̂ * (1 - p̂) * ((1/n1) + (1/n2))]
* where n1 = number of male customers, n2 = number of female customers
* SE = √[0.55 * (1 - 0.55) * ((1/130) + (1/70))]
* SE ≈ 0.0628
**5. Determine Critical Value**
* For a 95% confidence interval, the critical value (z*) from the standard normal distribution is 1.96.
**6. Calculate Margin of Error**
* Margin of Error = z* * SE
* Margin of Error = 1.96 * 0.0628
* Margin of Error ≈ 0.1231
**7. Construct the 95% Confidence Interval**
* Lower Limit: (p̂1 - p̂2) - Margin of Error = 0.0109 - 0.1231 = -0.1122
* Upper Limit: (p̂1 - p̂2) + Margin of Error = 0.0109 + 0.1231 = 0.1340
**Therefore, the 95% confidence interval for the difference between the proportions of male and female customers who prefer Coke Classic over Pepsi Cola is (-0.1122, 0.1340).**
**Interpretation:**
* Since the confidence interval includes zero, we cannot conclude with 95% confidence that there is a statistically significant difference between the proportions of male and female customers who prefer Coke Classic over Pepsi Cola in Chicago.
* It's possible that the observed difference in proportions is due to random chance.
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