In this section we will discuss inference for proportions (or percentages) such as the percentage of people who prefer Coke over Pepsi, who will vote PNP in the next election, who earn more than $50,000 per year etc. Note Most often problems are stated in terms of percentages instead of proportions but all the methods use proportions. Note Sometimes problems are for probabilites, that is the same as proportion in this context. All the methods discussed in this section are what we call large sample methods, that is they require a certain minimal sample size to work. Alternatively we can use MINITAB without the "assume normal approximation", then there are no assumptions! The website of the Puerto Rico Tourism Company has the results of a survey of Puerto Ricans and their vacation travel. 10% of the respondents had made a trip to Cabo Rojo, the highest number of any place in PR. Test at the 5% level of significance whether 5067 heads in 10000 flips are compatible with a fair coin. If indeed his coin had a probability of Heads of 0.505, how often would he have to flip his coin to have a power of 95%?
From the graph above you see that for proportions close to 0 or 1 a much smaller sample size is enough. If you use the formula for the mean although your problem is about a proportion you are guaranteed 0 points.
Sampling distribution of sample proportion The sampling distribution of a sample proportion is approximately normal (normal approximation of a binomial distribution) when the sample size is large enough. Large-sample confidence interval for p Use this method when the number of successes and the number of failures are both at least 15.
Let’s calculate a 90% confidence interval for the population proportion of arthritis patients who suffer some “adverse symptoms.” What is the sample proportion ? Because we have to use an estimate of p to compute the margin of error, confidence intervals for a population proportion are not very accurate.
Significance test for p The sampling distribution for is approximately normal for large sample sizes and its shape depends solely on p and n. P-values and one or two sided hypotheses — reminder And as always, if the p-value is as small or smaller than the significance level ?, then the difference is statistically significant and we reject H 0. A national survey by the National Institute for Occupational Safety and Health on restaurant employees found that 75% said that work stress had a negative impact on their personal lives. Software gives you summary data (sample size and proportion) as well as the actual p-value. Sample size for a desired margin of error You may need to choose a sample size large enough to achieve a specified margin of error. Comparing two independent samples We often need to compare two treatments used on independent samples.


Large-sample CI for two proportions For two independent SRSs of sizes n 1 and n 2 with sample proportion of successes 1 and 2 respectively, an approximate level C confidence interval for p 1 – p 2 is Use this method only when the populations are at least 10 times larger than the samples and the number of successes and the number of failures are each at least 10 in each samples. Cholesterol and heart attacks How much does the cholesterol-lowering drug Gemfibrozil help reduce the risk of heart attack?
Cholesterol and heart attacks Let’s now calculate the “plus four” CI for the difference in percentage of middle-aged men who suffer a heart attack (placebo – drug). If the null hypothesis is true, then we can rely on the properties of the sampling distribution to estimate the probability of drawing 2 samples with proportions 1 and 2 at random. Relative risk Another way to compare two proportions is to study the ratio of the two proportions, which is often called the relative risk (RR).
Objectives (BPS chapter 20) Inference for a population proportion ? The sample proportion ? The sampling distribution of ? Large sample confidence interval. From the Data at Hand to the World at Large Chapter 19 Confidence Intervals for an Unknown Population p Estimation of a population parameter: Estimating. Chapters 20, 21 Testing Hypotheses about Proportions Part II: Significance Levels, Type I and Type II Errors, Power 1.
The object of interest here is π, the proportion of people in the whole population who prefer Coke over Pepsi.
The study measures short trip leisure travel habits of average Puerto Rican families and allows for the monitoring of consumer preferences on a continuous basis. You want to find a 95% confidence interval for the percentage of voters for the PNP, with an arror of E=0.03. Find a 95% confidence interval for the population percentage of Puerto Ricans with type O+ blood.
Test at the 5% level whether the this shows that the percentage of type O+ blood is statistically significantly higher than the world percentage.
An experiment on the side effects of pain relievers examined arthritis patients to find the proportion of patients who suffer side effects. What is the sampling distribution for the proportion of arthritis patients with adverse symptoms for samples of 440?
We act as if we had four additional observations, two being successes and two being failures. You investigate a restaurant chain to see if the proportion of all their employees negatively affected by work stress differs from the national proportion p 0 = 0.75.
However, because the sampling distribution of is a function of the population proportion p, this process requires that you guess a likely value for p: p*. However, since the drug has been approved for sale over the counter, we can safely assume that no more than 10% of patients should suffer “adverse symptoms” (a better guess than 50%). We compare the incidence of heart attack over a 5-year period for two random samples of middle-aged men taking either the drug or a placebo.
We act as if we had four additional observations: one success and one failure in each of the two samples.


Test of significance This test is appropriate when the populations are at least 10 times as large as the samples and all counts are at least 5 (number of successes and number of failures in each sample). Patients would swallow a deflated balloon with tubes, and a cold liquid would be pumped for an hour to cool the stomach and reduce acid production, thus relieving ulcer pain. Obviously the proportion in the sample who prefer Coke over Pepsi will be our point estimate of π. A survey of 17,096 students in US four-year colleges collected information on drinking behavior and alcohol-related problems. This ensures that the standard deviation of is close to 3.The sample size n is large enough that the sampling distribution can be approximated with a normal distribution. For an SRS of size n drawn from a large population, and with sample proportion calculated from the data, an approximate level C confidence interval for p is: C is the area under the standard normal curve between ?z* and z*. If H 0 is true, the sampling distribution is known ? The likelihood of our sample proportion given the null hypothesis depends on how far from p 0 our is in units of standard deviation.
The smaller the p-value, the stronger the evidence against the null hypothesis and the more confident you can be about your interpretation. The treatment was shown to be safe, significantly reducing ulcer pain and widely used for years. The procedure for calculating confidence intervals for relative risk is more complicated (use software) but still based on the same principles that we have studied. How large a sample size is required depends in part on the value of p and the test conducted. This is valid when both expected counts—expected successes np 0 and expected failures n(1 ? p 0 )—are each 10 or larger. The magnitude or size of an effect relates to the real-life relevance of the phenomenon uncovered.
A randomized comparative experiment later compared the outcome of gastric freezing with that of a placebo: 28 of the 82 patients subjected to gastric freezing improved, while 30 of the 78 in the control group improved. The age at which a woman gets her first child may be an important factor in the risk of later developing breast cancer.
The researchers defined "frequent binge drinking" as having five or more drinks in a row three or more times in the past two weeks. An international study selected women with at least one birth and recorded if they had breast cancer or not and whether they had their first child before their 30 th birthday or after. However, magnitude is not necessarily equivalent to how theoretically or practically relevant an effect is.



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