Express the confidence interval (0.024, 0.112) in the form of p-hat – E < p < p-hat + E = __< p < __
I don't understand how to perform this formula. Please help I am about to graduate and this is my last class, and I am struggling as I only know basic Math! Please can some one help. Thank you
I'm sorry to hear that you're struggling with this concept. I'd be happy to help you understand how to express a confidence interval in the form of p-hat – E < p < p-hat + E.
Let's break it down step by step.
1. First, we need to understand what p-hat represents. In statistics, p-hat is the sample proportion, which is an estimate of the true population proportion denoted by p. It is calculated by dividing the number of successes (or "favorable outcomes") by the total number of observations in the sample.
2. The confidence interval expresses the range of values within which we have a certain level of confidence (usually represented as a percentage) that the true population proportion lies. In this case, the confidence interval you provided is (0.024, 0.112), which means that we are 95% confident that the true population proportion (p) lies within this range.
3. To express the confidence interval in the form p-hat – E < p < p-hat + E, we need to calculate the margin of error (E). The margin of error represents the maximum likely difference between the sample proportion (p-hat) and the true population proportion (p).
4. To calculate the margin of error, we subtract the lower bound of the confidence interval (0.024) from the sample proportion (p-hat) and then divide the result by 2:
E = (p-hat - lower bound) / 2
E = (p-hat - 0.024) / 2
5. To calculate the lower bound of the confidence interval (p-hat – E), we subtract the margin of error (E) from the sample proportion (p-hat):
p-hat - E = p-hat - (p-hat - 0.024) / 2
6. To calculate the upper bound of the confidence interval (p-hat + E), we add the margin of error (E) to the sample proportion (p-hat):
p-hat + E = p-hat + (p-hat - 0.024) / 2
7. Finally, we substitute the calculated values into the expression p-hat – E < p < p-hat + E to obtain the desired form of the confidence interval:
0.024 < p < 0.112
I hope this explanation helps you understand how to express a confidence interval in the form p-hat – E < p < p-hat + E. If you have any further questions, please feel free to ask. Good luck with your studies!