Population Proportions The model used is the Binomial Distribution. p represents the proportion of successes in the population. represents the proportion of successes in our sample of size n. That is

Hypothesis testing of a population proportion H 0 : p = p 0 – Ha: p < p 0 or – Ha: p > p 0 or (discouraged) – Ha: p ≠ p 0

Normal Approximation to the Binomial Before computers, p-values were calculated using the normal approximation to the binomial. Recall, for large n (generally 30 or larger), the binomial distribution is approximately normal, with mean  = n*p and standard deviation sqrt(n*p*(1-p)). However, we have computers! Just use the binomial Excel worksheet.

Example The strength of a pesticide may be measured by the proportion of pests that it kills. One pesticide advertises that if used properly it kills at least 99% of the pests. To see if the advertised figure is correct, 1000 pests are exposed to the pesticide, following the directions carefully, and 980 of them are killed. Is there sufficient evidence to conclude that the advertised figure of 99% is too high?

Solution H0: p =.99Ha: p <.99 The p-value of which almost certainly means the manufacturer has overstated their claims. N =1000 prop =0.99 least # successes =0 highest # successes =980 prob =0.0033

Significance Level and p-value Recall that (1-confidence level) = α specifies an area in the tail of the distribution which constitutes the rejection region for the null hypothesis. The p-value represents the probability that our result can be explained just by the fact that samples vary. We reject the null hypothesis when the p-value is less than α. Concluding our result cannot be explained simply because samples vary.

Possible Errors Truth Our decision H 0 TrueH 0 False Fail to reject H 0 Not an errorType II error Reject H 0 Type I errorNot an error

Estimating a population proportion Starting with a random sample of size n, and the sample proportion we form an interval of the form

Example Suppose an opinion poll predicted that, if the election were held today, the Conservative party would win 60% of the vote. The pollster might attach a 95% confidence level to the interval 60% plus or minus 3%. That is, he thinks it very likely that the Conservative party would get between 57% and 63% of the total vote.

Confidence intervals Confidence intervals are calculated using the formula

Margin Of Error Notice that the width of the interval is From looking at the graph below we can see that the maximum of p*(1-p) occurs when p =0.5

Sample size for given margin of error n > (z*/m)^2 * (0.25)

Example If we want to be certain that a poll will have a margin of error of no more than 5%, at the 90% significance level we calculate and conclude that if our sample size is larger than 271 or margin of error will be less than 5%.