# 3/1/2023

## Discussion of CLT for Sample Proportion

## Statistical Inference

- Point Estimation
- Confidence Intervals: point estimator ± margin of error

Definition: A 100*C% confidence interval (C.I.) for a parameter (say theta) is given by two statistics L and U such that P(L<= theta <= U) = C.

- Derivation of a CI for one proportion

## In class exercises

- Suppose that X~B(1000, .5)
- Find the probability that X is at most 532.
- Find the mean and standard deviation of X.
- Approximate the probability that X is at most 532 using the CLT. Are you happy with the accuracy of your approximation?
- Now, use the continuity correction and add 0.5 to 532 and use the CLT again. Did the accuracy of your approximation improve?

- Suppose that X~B(30, .6)
- Find the probability that X is at most 25.
- Find the mean and standard deviation of X.
- Approximate the probability that X is at most 25 using the CLT. Are you happy with the accuracy of your approximation?
- Now, use the continuity correction and add 0.5 to 25 and use the CLT again. Did the accuracy of your approximation improve?

- Suppose that the Office of Admissions has a target of 415 students and that they admit 1400 students. Previous experience at this institution indicates that 27% of the students who are offered admission will accept the offer. That is, let X denote the number of students who accept the offer of admission to this college, and use X ~ B(1400, .27).
- Find the probability that more than 415 students accept the offer.
- Find the mean and standard deviation of X.
- Approximate the probability that more than 415 students accept the offer using the CLT. Are you happy with the accuracy of your approximation?
- Now, use the continuity correction and use the CLT again. Did the accuracy of your approximation improve?

- Textbook exercises - 5.3, 5.5, 5.7, and 5.9

## Please read Section 5.3 for class on Friday.

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