Lecture12.chapt8

Lecture12.chapt8 - Lecture 12, Section 8.1 & 8.2...

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Proportions Many statistical studies produce counts rather than measurements. Example: Did you vote in the last election? The response would be either a “Yes” or a “No”. The variable is categorical, the response is the value the variable takes on for each unit/person. If I did a survey of this class, I could accumulate the count of “Yes” responses and describe this count as a proportion of the total. Example: What academic year are you in at Purdue. The response would be either “Freshman”, “Sophomore”, “Junior”, or “Senior”. Again, I could accumulate the count of each and describe each as a proportion of the total. Population and Sample proportions: In statistical sampling we often want to estimate the proportion, p, of “successes” in a population. “Success” is when the categorical variable takes on one particular value. p = count of successes in population size of population = X / N We take a sample of our population; our estimator is the sample proportion of successes: p = count of successes in sample size of sample = X / n Example: 1. You flip a coin 20 times and record whether a head or a tail is tossed. In this sample, a head is recorded 11 times. What is the sample proportion of heads? Page 1
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Inference for a Single Proportion: So far we have only looked at making statistical inference on population means, a measurement of some quantitative variable of interest. Now we will look at making statistical inference on a categorical variable using the proportion of some outcome/success in a population. Examples: How common is it for students at Purdue to fail a class? Out of a sample of 200 students, 50 of them have failed at least one class, or 25% . Based on these data, what can we say about all students at Purdue? What proportion of golfers in the USA have made at least one hole-in- one in their lifetime. From an SRS of 50 golfers 25 of them had made a hole-in-one. What can we say about all golfers in the USA? In both examples above we are interested in estimating the unknown proportion p from a population. The estimate of that population parameter p is the sample proportion µ p , a statistic. Sampling Distribution of a Sample Proportion: Choose an SRS of size n from a large population with population proportion p having some characteristic of interest. We normally call whatever characteristic we are studying a “success.” Let X be the count of successes in the sample and µ p be the sample proportion of success, µ p = X/ n Also: The sampling distribution of µ p is approximately normal for a SRS from a large population and is closer to a normal distribution when the
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Lecture12.chapt8 - Lecture 12, Section 8.1 & 8.2...

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