Final answer:
To find the 99% confidence interval for the true proportion of children who like ice cream based on the sample, we need to calculate the standard error, find the critical value (z⋆), and construct the confidence interval.
Using the given information, the confidence interval is approximately 0.8551 to 0.9149.
Step-by-step explanation:
A confidence interval is a range of values that is likely to contain the true population proportion with a certain level of confidence. In this case, we want to find the 99% confidence interval for the true proportion of children who like ice cream based on the sample.
- To calculate the confidence interval, we need to determine the standard error of the proportion. The formula for the standard error is:
- Standard Error = sqrt((p_hat * (1-p_hat))/n)
Given that there are 885 out of 1000 children who like ice cream, the proportion estimate is 0.885.Plugging in the values into the formula, we have:
- Standard Error = sqrt((0.885 * (1 - 0.885))/1000)
- Standard Error = 0.0116
Next, we need to find the critical value (z⋆) for a 99% confidence level. We can use a standard normal distribution table or a calculator to find this value. For a 99% confidence level, z⋆ is approximately 2.58.Now, we can calculate the margin of error by multiplying the standard error by the critical value:
- Margin of Error = z⋆ * Standard Error
- Margin of Error = 2.58 * 0.0116
- Margin of Error = 0.0299
Finally, we can construct the confidence interval by subtracting and adding the margin of error to the proportion estimate:
- Lower Bound = 0.885 - 0.0299
- Lower Bound ≈ 0.8551
- Upper Bound = 0.885 + 0.0299
- Upper Bound ≈ 0.9149
Therefore, the 99% confidence interval for the true proportion of children who like ice cream is approximately 0.8551 to 0.9149.