1 Chapter 5 Sampling Distributions

2 The Distribution of a Sample Statistic Examples  Take random sample of students and compute average GPA in sample.  Ask 20 people whether or not they will vote NPA. Get number who say yes.  Get proportion of people in a SRS who favor the Olympics. These statistics ( in bold face) are random variables.  The results vary from sample to sample Probability distribution of a statistic is called its sampling distribution

3 Some important sampling distributions Distribution of  Sample average  Sample count (number of successes in sample)  Sample proportion (proportion of successes in sample)  Many others

4 Binomial setting for sample counts and proportions (Section 5.1) Fixed number, n, of observations. Only two outcomes for each observation – call them Success or Failure (S/F) Observations are all independent. Probability of success, p, is same for each trial. _____________________ In the above setting can look at the prob. dist. of sample counts of successes and at the sample proportion of successes

5 Binomial Distribution Let X be the number of successes in a binomial setting. The probability distribution of X is called the binomial distribution with parameters n and p.

6 Binomial Formula Loaded coin – each toss has prob 0.6 of coming up heads. Toss 3 times. Toss 10 times.

7 Binomial Example Six customers enter a restaurant. With probability 0.75, any given customer purchases a meal. Purchase are independent of one another. Analyze number of meals purchased.

8 Tables and Excel for Computing Binomial Probabilities Table C (not in my text, but should be there) has binomial probabilities  Gives prob. of exactly k successes in n trials, for a given p. Excel  BINOMDIST(6,20,0.4,FALSE) Prob. of exactly 6 successes in 20 trials with prob. of success 0.4 on each trial.  BINOMDIST(6,20,0.4,TRUE) Prob. of 6 or fewer successes in 20 trials with prob. of success 0.4 on each trial. Midterms – use binomial formula and Excel formula. HW – formula, table or Excel.

9 Example: 20% of govt. employees are dissatisfied with their wage. I take a SRS of 40 employees. What is the expected number and std. deviation of the number of dissatisfied employees in my sample?

10 Sample Proportion What about the proportion of dissatisfied employees in my sample? Called the sample proportion. Its probability can be computed via binomial dist. Also, sample proportion has  Mean=p  Std. Dev = SQRT[(p(1-p))/n].

11 Normal Approximation for Counts and Proportions Computing binomial probabilities when n is large.  E.g Suppose 90% of Canadians favor Liberal party! Gallup poll takes SRS of 1600 Canadians. How likely is it that at least 1460 of those sampled favor Liberals?  Approximately binomial but could spend forever computing this. What to do???

12 Normal approximation for counts and proportions Same thing for sample proportion.  Distribution of sample proportion is approximately normal with mean = p and std. dev = sqrt[p(1-p)/n]

13 Example of normal approximation to binomial Consider our Gallup poll. Have p =.9, n = Recall that we want prob. that at least 1460 of the people in our sample favor Liberals. We’ll also approximate prob. that at least 85% of people in our sample favor Liberals.

14 The continuity correction (optional ) Improves the accuracy of normal approximation to binomial. Can lead to substantially better approximations, especially for smaller values of n. Here it is: Adjust P(a <X< b) by using  P(a-0.5 < X <b+0.5), Adjust P(X > a) by using  P(X > a-0.5) Adjust P(X<b) by using  p(x<b+0.5) Try it for our Gallup poll example. Try for situation when p =.5, n = 25.

15 Review question on binomial (you do) 70% of all mortgage applicants are granted a loan. 5 people apply for a mortgage today. What is prob. exactly 3 get a loan? Suppose 100 people apply for a mortgage each month.  What is mean and std dev of # who are granted loans?  How likely is it that at least 75 of the 100 people are granted loans?  Find the 95 th percentile of the number who are granted loans. Hints For second part, use normal approx to binomial Mu = np Sigma = sqrt[n*p*(1-p)] Then it’s just a normal distribution question.

16 Sampling Distribution of a Sample Mean Section 5.2

17 Sample mean as a random variable Suppose I want to estimate the mean annual return on TSE stocks over the past year. I take a random sample of 20 stocks and get the average return for past year in the sample. That’s my estimate of the true mean return for TSE But different samples would give different estimates (results vary from sample to sample). Thus the sample average ( also known as the sample mean) is a random variable.

18 Distribution of Sample Means Sample averages are less variable than individual observations Averages from big samples should be less variable than averages taken from small samples. It turns out that the sample averages tend to follow a normal distribution (more on this later)

19 Example:  I take a SRS of n= 36 people from a large population with mean 50 and std deviation 24.  What is the mean and standard deviation of the sampling distribution of xbar?

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21 Example – Sampling Dist. Of Sample Mean The fills from a vending machine are normally distributed with a mean of 250 ml and a std. dev of 15 ml. I take a sample of n = 100 fills from the machine. How likely is it that the average fill from the 100 cups is between 247 and 253 ml.?

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23 Example – An Application of Central Limit Theorem: The distribution of the time required to complete a certain task is highly skewed to the right. However, we know that the mean time required is 60 minutes, with a standard deviation of 30 minutes. Consider a SRS of 225 people who complete the task. Estimate the probability that at the average time required for these people to complete the task lies between 57 and 63 minutes.

24 Weighted sums and differences of normal variables are normally distributed Accountants incomes are normally distributed with a mean of $100K and a standard deviation of 20K. Lawyers incomes are normally distributed with a mean of $80K and a standard deviation of 40K. Incomes of lawyers and accountants are independent. I select an accountant and a lawyer at random. How likely is it that their combined income exceeds $220K? How likely is it that the accountant makes more than the lawyer?