b. A mobile company XYZ wants to conduct a survey to find the proportion P of those customers that are happy with the purchase of their new mobile phone. The company took a random sample of 1000 customers who recently purchased the new mobile phone. They found that among those, 400 customers were happy with their purchase. Find the 95% Confidence Interval for P?
To find the 95% confidence interval for the proportion P of customers who are happy with their purchase, we can use the formula:
Confidence Interval = sample proportion ± margin of error
1. Calculate the sample proportion:
The sample proportion is calculated by dividing the number of customers who are happy with their purchase (400) by the total number of customers in the sample (1000).
Sample Proportion (p̂) = 400 / 1000 = 0.4
2. Calculate the margin of error:
The margin of error depends on the significance level (alpha), which is (1 - confidence level). Since we want a 95% confidence interval, our alpha will be 0.05. We can then use the formula:
Margin of Error = Z * √(p̂ * (1 - p̂) / n)
where Z is the z-score for the desired confidence level, p̂ is the sample proportion, and n is the sample size.
The z-score corresponding to a 95% confidence level is approximately 1.96 (from the standard normal distribution table).
Margin of Error = 1.96 * √(0.4 * (1 - 0.4) / 1000)
3. Calculate the confidence interval:
Confidence Interval = Sample Proportion ± Margin of Error
Confidence Interval = 0.4 ± Margin of Error
Substitute the value of the margin of error into the equation:
Confidence Interval = 0.4 ± 1.96 * √(0.4 * (1 - 0.4) / 1000)
Now you can calculate the confidence interval by calculating both the lower and upper bounds:
Lower Bound = 0.4 - 1.96 * √(0.4 * (1 - 0.4) / 1000)
Upper Bound = 0.4 + 1.96 * √(0.4 * (1 - 0.4) / 1000)
The 95% confidence interval for the proportion P represents the range within which we can be 95% confident that the true proportion of happy customers falls.