# Randomness and Probability

Sampling Distributions What is a Sampling Distribution? Section 7.1 Reference Text: The Practice of Statistics, Fourth Edition. Starnes, Yates, Moore Objectives 1. Parameter and Statistic Lets define some variables

2. Sampling variability 3 different types of distributions to distinguish from. Wording! Be careful, some AP errors 3. Describing sampling distributions Unbiased estimators 4. Why the size of our sample matters 5. Variability and Bias

Statistic and Parameter As we begin to use sampling data to draw conclusions about wider population, we must be clear about whether a number describes a sample or a population: If we are talking about a PopulationParameter If we are talking about a Sample . Statistic Parameter & Statistic A Parameter is a number that describes some characteristic of the population. In statistical practice,

the value of a parameter is usually not known because we cannot examine the entire population Makes sense! Thats why we take samples! A Statistic is a number that describes some characteristic of a sample. The value of a statistics can be computed directly from the sample data. We often use a statistics to estimate an unknown parameter. Statistic..Sample ParameterPopulation

Variables To Distinguish From! Parameter p (fraction, decimal, percent) NOT usually known! Statistic (fraction, decimal, percent)

Example For the following example identify the population, parameter, sample, and statistic The Gallop Poll asked a random sample of 515 U.S adults whether or not they believe in ghosts. Of the respondents, 160 said yes. Population? All U.S Adults

Parameter? p, the proportion of all U.S adults who believe in ghosts. Sample? 515 people Statistic? =.31 Check Your Understanding

Each boldface number in Question 1 and 2 is the value of either a parameter or a statistic. State which is whichuse correct vocabulary 1. On Tuesday the bottle of Arizona Ice Tea filled in a plant were supposed to contain an average of 20 ounces of iced tea. Quality control inspectors sampled 50 bottles at random from the days production. These bottles contained an average of 19.6 ounces of iced tea. Parameter = = 20 ounces, Statistic = ounces 2. On a New-York-To-Denver flight, 8% of the 125 passengers were selected for random security screening before boarding.

According to the Transportation Security Administration (TSA), 10% of passengers at this airport are chosen for random screening. Parameter = = .10, 10% of passengers Statistic = Sampling Variability Sampling Variability Sample

Population Sample Sample Sample Sample Sample Sample

Sample ? Sampling Distribution Sampling distribution of a statistic is the distribution of values taken by the statistic in all possible samples of the same size from the same population. Q: what would happen if we took many samples A: -Take a large number of samples from the same population

- Calculate the statistic for each sample - Make a graph of the values of the statistic - Examine the distribution displayed in the graph for shape, center, and spread, as well as outliers (Hey Look! C.U.S.S.) C.U.S.S Note: small change when talking about sampling distributions

C- center, this is your mean. U- unusual points, any notable outliers? S- spread, this is your standard deviation S- Shape, is it symmetrical? Single peak? AP Exam Tip Terminology matters. Dont say sample distribution when you mean sampling

distribution. You will lose credit on free response questions for misusing statistical terms. Yeap, it really is the difference of just ing that determines full credit or docked points Activity! Suppose I have 200 chips in my Finding Dory Bin, 100 are red and the remaining are blue the population parameter of red chips is thus Directions: Grab 20 random chips and record

the sample proportion of red chips. Then put them back in the bag. Record your on the board. We will have 28 (A day) 11 (B-day) samples to create our sampling distribution! Population Distributions vs. Sampling Distributions There are actually three distinct distributions involved when we sample repeatedly and measure a

variable of interest. 1) The population distribution gives the values of the variable for all the individuals in the population. 2) The distribution of sample data shows the values of the variable for all the individuals in the sample. 3) The sampling distribution shows the statistic values from all the possible samples of the same size from the population.

Sampling Distribution Unbiased Estimator Note: unbiased doesnt mean perfect An unbiased estimator will almost always provide an

estimate that is not equal to the value of the population parameter. It is called unbiased because in repeated samples, the estimates wont consistently be too high or consistently too low. We also assume that the sampling process we are using has no bias. No sampling or non-sampling error present, just sampling variability. If they did exist, then that would lead to estimates too low or too high. Unbiased Estimator

Assume that the true population proportion is 60. Q: Which graph is an unbiased estimator? Unbiased Estimator Assume that the true population proportion is 60. Q: Which graph is a biased estimator? Check Your Understanding Mars, Inc, says that the mix of colors in its M&Ms Milk Chocolate Candies is 24% blue, 20% orange, 16% green, 14% yellow, 13% red, and 13% brown. Assume that the companys claim is true. We want to examine the

proportion of orange M&Ms in repeated random samples of 50 candies. 1) Which of the graphs that follow could be the approximate sampling distribution of the statistic? Explain your choice. Check Your Understanding 1) Which of the graphs that follow could be the approximate sampling distribution of the statistic? Explain your choice. Why Size Matters Television executives and companies who

advertise on TV are interested in how many viewers watch particular shows. According to Nielsen ratings, Survivor was one of the mostwatched television shows in the United States during every week that it aired. Suppose that the true proportion of U.S adults who have watched Survivor is p =0.37 Lets look at an illustration that clearly shows why size makes a world of difference Why Size Matters Figure 7.7 (a) shows the results of drawing 1000 SRSs of size

n= 100 from a population with p = 0.37. Figure 7.7 (b) shows the results of drawing 1000 SRSs of size n= 1000 from a population with p = 0.37. Both graphs are drawn on the same horizontal scale to make comparison easier: n=100 n=1000 What Does Size Do In Statistics?

There is a clear advantage to larger samples. They are much more likely to produce an estimate close to the true value of the parameter. Said another way, larger random samples give us more precise estimates than smaller random samples. This leads us to the Variability of a Statistic Variability Of a Statistic The variability of a statistic is described by the spread of its sampling distribution. This spread is

determined primarily by the size of the random sample. Larger samples give smaller spread. The spread of the sampling distribution does not depend on the size of the population, as long as the population is at least 10 times larger than the sample. Taking larger sample doesnt fix bias. Remember that even a very large voluntary response sample or convenience sample is worthless because of bias. Describing

Sampling Distributions Bias, variability, and shape We can think of the true value of the population parameter as the bulls- eye on a target and of the sample statistic as an arrow fired at the target. Both bias and variability describe what happens when we take many shots at the target. Bias means that our aim is off and we consistently miss the bulls-eye in the same direction. Our

sample values do not center on the population value. High variability means that repeated shots are widely scattered on the target. Repeated samples do not give very similar results. The lesson about center and spread is clear: given a choice of statistics to estimate an unknown parameter, choose one with no or low bias and

minimum variability. Bias, Variability, and Shape Ideally, wed like our estimates to be accurate (unbiased) and precise (have low variability) Objectives 1. Parameter and Statistic Lets define some variables

2. Sampling variability 3 different types of distributions to distinguish from. Wording! Be careful, some AP errors 3. Describing sampling distributions Unbiased estimators 4. Why the size of our sample matters 5. Variability and Bias Homework

7.1 Homework Worksheet Start Chapter 7 Reading Guide

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