Stats 11 (Fall 2004) Lecture Note Instructor: Hongquan XuIntroduction to Statistical Methods for Business and EconomicsChapter 7 Sampling Distributions of EstimatesWe have studied probability, we have discussed random variables and their probability distributions. Nowwe turn to Statistical Inference: process of using data to draw conclusions about some wider population.Section 7.1 Parameters and EstimatesSome distinctions to keep in mind ...• Population versus Sample• Parameter versus Statistic and EstimateDefinitions:• Parameter: A number that describes the population.• Statistic (or estimator): An abstraction that describes a sample.• Estimate: A number calculated from the data to estimate an unknown parameterNotation:• Population proportion p versus sample proportion!P or !p• Population mean µ versus sample mean X or xSince we hardly ever know the true population parameter value, we take a sample X1, X2, ..., Xnand use thesample statistic to estimate the parameter. When we do this, the sample statistic may not be equal to thepopulation parameter, in fact, it would change every time we take a new sample (that is why we call thisprocess ESTIMATION ). Will the observed sample statistic value be a reasonable estimate? If the sampleis a random sample, then we will be able to say something about the accuracy of the es timation process.What do we mean by a random sample?Recall that X1, X2, ..., Xnis a random sample if ...1.2.1Example:A poll was conducted by The Heldrich Center for Workforce Development (at Rutgers University).A random sample of 1000 workers resulted in 460 (for 46%) stating they work more than 40 hours per week.Identi fy:Population =Parameter =Sample =Estimate =Can anyone say how close this observed sample proportion !p is to the true population proportion p? If wewere to take another random sample of the same size n = 1000, would we get the same value for the sampleprop ortion !p ?The value of a statistic from a random sample, like !p , will vary from sample to sample. So a statistic(also called estimator) is a random variable and it will have a probability distribution. This probabilitydistribution is called the sampling distribution of the statistic. We will study in this chapter the samplingdistribution of the following statistics:Sample Mean: X = the mean of a sample of size nSample Prop ortion:!P =Xn= the proportion of successes in a sample of size nSection 7.2 Sampling Distribution of the Sample MeanScenario:A poll was conducted by The Heldrich Center for Workforce Development (at Rutgers University).A sample of 1000 workers resulted in a mean number of hours worked per week of 43.Population =Parameter =Sample =Estimate =Can anyone say how close this observed sample mean x is to the true population mean µ?If we were to take another random sample of the same size n = 1000, would we get the same value for thesample mean x?Picture Model:2The value of the sample mean from a random s ample will vary from sample to sample. So the sample mean(a statistic) is a random variable and it will have a probability distribution. This probability distribution iscalled the sampling distribution of the statistic. In Section 7.2 we learn about the sampling distributionfor a sample mean, X = the mean of a sample of size n. What can we say about the X basket (population)?Notation:• population mean µ• population standard deviation σ• random sample of size n: X1, X2, . . . , Xn• sample meanX =X1+X2+...+XnnMain Results:1. The average of all of the possible sample mean values(from all possible samples of the same size n) is equal toThat is, the mean of the sample means is equal to the population mean.Thus the sample mean is an estimator of the population mean.2. The standard deviation of all of the possible sample mean values is equal to the original populationstandard deviation divided by√n.This is approximately the average distance of the possible sample mean values from the popula-tion mean. It is a measure of the accuracy of the process of using a sample mean to estimate the p opulationmean.Would this quantityσ√ntell me how far away a particular x value is from µ?Note: If the sample size increases, the standard deviation decreases, which says the possible sample meanvalues will be closer to the true population mean (on average).3. If the population has a normal distribution,then the distribution of the sample mean is3Note: Sums and differences of independent normal r.v. also have a normal distribution (Section 6.4.3).4. If the population is not necessarily normally distributed,but the sample size n is large,then the distribution of the sample mean isThis result is called theSee Figures 7.2.3- 7.2.5 on pages 285-287.Rule of Thumb: In general, the approximation is good if4ExampleACT scores: Normally distributed with mean of 18.6 and standard deviation of 5.9.a. What is the probability that a single student randomly selected will score 21 or higher?b. A random sample of 50 students is obtained. What is the probability that the mean score for these 50students will be 21 or higher?c. What if the normal distribution assumption were not given? How would your answer to part b change?Explain.Example: Uniform DistributionSupp ose the random variable X = actual flight time (in minutes) for Delta Airlines flights from Cincinnatito Tampa follows a uniform distribution over the range of 110 minutes to 130 minutes.a. Sketch the distribution for X (include labels for the axes and some values on the axes).b. Suppose we were to repeatedly take a random sample of size 100 from this distribution and c ompute thesample mean for each sample. What would the histogram of the sample mean values look like? Provide asketch and any details about the distribution of the sample mean that you can.5Example:Grinding Machine prepares axles: target diameter = 40.125 mm. The standard deviation of diameters is0.002 mm. A (random) sample of 4 axles taken each hour and the sample mean diameter is recorded andplotted. What will be the mean and standard deviation of the numbers recorded?Note:When n = 1 we have:When n = 4 we have:So to reduce the standard deviation in half, we mustQuestion: What sample size would be needed for the standard deviation of the sample mean to be 0.0005mm?Standard Error of Sample MeanRecall that the s.d. of sample mean X isIn practice, we don’t know s.d. σ, so we have to estimate σ and sd(X).We can estimate σ byand estimate sd(X) byThe Standard Error of the