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?

Question

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?

Answer 1

To find the 95% confidence interval for the proportion P, we can use the formula:

Confidence Interval = p̂ ± Z * √(p̂(1-p̂)/n)

where:
- p̂ (pronounced "p-hat") is the sample proportion
- Z is the critical value from the standard normal distribution corresponding to the desired confidence level (95% confidence level corresponds to Z = 1.96)
- n is the sample size

In this case, p̂ is the proportion of customers who are happy with their purchase, which is 400/1000 = 0.4. The sample size, n, is 1000.

Now, let's substitute these values into the formula:

Confidence Interval = 0.4 ± 1.96 * √(0.4(1-0.4)/1000)

Simplifying the expression:

Confidence Interval = 0.4 ± 1.96 * √(0.24/1000)

Confidence Interval = 0.4 ± 1.96 * 0.01549

Now, we can calculate the lower and upper bounds of the confidence interval:

Lower bound = 0.4 - 1.96 * 0.01549
Upper bound = 0.4 + 1.96 * 0.01549

Therefore, the 95% confidence interval for the proportion P is approximately:

0.4 ± 0.0303

Taking the lower and upper bounds into account, the confidence interval is (0.3697, 0.4303). This means that we are 95% confident that the proportion of all customers who are happy with their purchase falls between 36.97% and 43.03%.