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 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
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) =
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 Note that the d.f. is 24 (n=25)
We can now construct the confidence interval: or