simple random sample of 50 women on new product of the 50 women, 60% reported they like the new product, fina a 90% confidence interval for the true proportion of women.
To find the 90% confidence interval for the true proportion of women who like the new product based on your sample data, you can use the formula:
CI = p̂ ± z * √(p̂ * (1 - p̂) / n)
where:
- CI is the confidence interval
- p̂ is the sample proportion (in this case, the proportion of women who like the new product, which is 60% or 0.60)
- z is the z-score corresponding to the desired level of confidence (90% confidence corresponds to a z-score of approximately 1.645 for a large sample size)
- n is the sample size (in this case, 50)
Let's plug the values into the formula and calculate the confidence interval.
p̂ = 0.60
z = 1.645
n = 50
CI = 0.60 ± 1.645 * √(0.60 * (1 - 0.60) / 50)
Now, let's calculate the values:
p̂ * (1 - p̂) = 0.60 * (0.40) = 0.24
√(0.24 / 50) ≈ 0.098
CI = 0.60 ± 1.645 * 0.098
To calculate the upper and lower bounds of the confidence interval, perform the addition and subtraction:
Upper bound = 0.60 + 1.645 * 0.098
Lower bound = 0.60 - 1.645 * 0.098
Finally, we can calculate the values:
Upper bound ≈ 0.60 + 0.162
Upper bound ≈ 0.762
Lower bound ≈ 0.60 - 0.162
Lower bound ≈ 0.438
Therefore, the 90% confidence interval for the true proportion of women who like the new product is approximately 0.438 to 0.762.