Confidence Intervals for Proportions and Variances

1. Confidence Intervals for Proportions and Variances QSCI 381 – Lecture 23 (Larson and Farber, Sects 6.3 and 6.4)

2. Point Estimate for a Proportion • The probability of success in a single trial of a binomial experiment is p. This probability is a population proportion. • The point estimate for p, the population proportion of success, is the proportion of successes in the sample, i.e.: Note: is referred to as p-hat.

3. Example • We measure 50 fish, 34 of them have evidence of a parasite. The estimate of the population proportion that have the parasite is:

4. Confidence Interval for a Proportion • We can calculate approximate confidence intervals for an estimate of a proportion using the normal approximation to the binomial distribution, i.e.: • A for the population proportion p is: where: c-confidence interval

5. Example • Find a 90% confidence interval for the proportion of our fish population that has the parasite: • Identify n, and . • Check whether the binomial distribution can be approximated by a normal distribution, i.e. • Determine the critical value . • Calculate the maximum error of estimate. • Construct the c-confidence interval.

6. Minimum Sample Size to Estimate p • Given a c-confidence level and a maximum error of estimate E, the minimum sample size n needed to estimate p is: • This formula depends on and which are the quantities were are trying to estimate. Either set the values for these quantities to preliminary estimates or set . Why is this latter assumption “conservative”?

7. Example • You wish to sample a population and you want to estimate, with 95% confidence, the proportion that are mature to within 0.01. How large must the sample size be?

8. Point Estimate for a Variance • The point estimate for 2 is s2 and the point estimate for is s. s2 is the most unbiased estimate of 2.

9. The Chi-square Distribution-I • If the random variable X has a normal distribution, then the distribution of forms a for samples of any size n>1. chi-square distribution

10. The Chi-square Distribution-II • The properties of the chi-square distribution are: • All values of 2 are greater than or equal to zero. • The area under the chi-square distribution equals one. • Chi-square distributions are positively skewed. • The chi-square distribution is a family of curves, each determined by the degrees of freedom. To form a confidence interval for 2, use the chi-square distribution with degrees of freedom equal to one less than the sample size, i.e. d.f.=n-1.

11. The Chi-square Distribution-III

12. The Chi-square Distribution-IV (1-c)/2 Chi-square distribution with 10 degrees of freedom Note: the distribution is not symmetric. (1-c)/2

13. The Chi-square Distribution-V CHIINV(p,d.f.) Chiinv(0.05,10) = 18.307 Chiinv(0.95,10) = 3.940

14. Confidence Intervals for 2 and  • A c-confidence interval for a population variance and standard deviation is:

15. Example • The density of a fish species is estimated by taking 25 samples. The sample standard deviation is 10 kg / ha. Construct a 95% confidence interval for the population standard deviation. • We first find the critical chi-square values. • We want a 95% confidence interval so the probability below the left limit and above the right limit should be 0.025. Note that the d.f. is 24 (n=25) • We can now construct the confidence interval: or