Population, Sampling, and Statistical Inference Notes

7.1 ❖ Population - a group of objects or people we wish to study. ➢ Example : the group of all U.S. citizens, or all U.S. citizens between the ages of 13 and 18, or all senior citizens. ❖ Parameter - a numerical value that characterizes some aspect of this population . ➢ Im other words, a numerical fact about a population is a parameter. (parameter measures the population; such as the average height of everyone in a school). ➢ Example : political pollsters want to know what percentage of people say they will vote in the next election. ✓ Parameters are represented by Greek characters ❖ Two frequently used parameters: the mean of a population and the population proportion. ❖ Census- a survey in which every member of the population is measured. ➢ Example : if you wish to know the percentage of people in your classroom who are left-handed, you can perform a census. The classroom is the population and the parameter is the percentage of left-handers. ➢ Sometimes we try to take a census with a large population (such as the U.S. Census), but it can be too expensive and can be filled with complications caused by attempting to track down and count people who may not want to be found. ■ Example : U.S. Census tends to undercount poor, urban-dwelling residents, as well as undocumented immigrants. ❖ Sample - a collection of people or objects taken from the population of interest. ➢ Once a sample is collected, we measure the characteristic we’re interested in. ❖ Statistic - a numerical characteristic of a sample of data. ➢ In other words, a numerical fact about a sample is a statistic. (statistics measures the sample; such as the average height of 5 people randomly selected in the same school). ➢ We use statistics to estimate parameters. ■ Example : We might be interested in knowing what proportion of all registered voters will vote in the next national election. The proportion of registered voters will vote in the next election is our parameter. Our method to estimate this parameter is to survey a small sample. The proportion of the sample who say they will vote in the next election is a statistic . ✓ Statistics are represented by English letters. ➢ Statistics are sometimes called estimators , and the numbers that result are called estimates . ■ Example : Our estimator is the proportion of people in a sample who say they will vote in the next election. The number 0.75 from the sample of people who say that they will vote is the estimate ❖ KEYPOINT : A statistic is a number that is based on data and used to estimate the value of a characteristic of the population. Thus it is sometimes called an estimator. ❖ Statistic inference : the art and science of drawing conclusions about a population on the basis of observing only a small subset of the population. Example 1: Pew Poll: Age and the Internet In February 2014 (about the time of Valentine’s Day), the Pew Research Center surveyed 1428 adults in the U.S. who were married or in a committed partnership. The survey found about 25% of cell phone owners felt that their spouse or partner was distracted by his or her cell phone when they were together. Question : Identify the population and the sample. What is the parameter of interest? What is the statistic? Solution : Population Sample All American adults in the U.S. who were married or in a committed partnership and owned a cell phone. Parameter of Interest 1428 adults in the U.S. The percentage of all who were married or in adults in the U.S. who a committed were married or in a partnership and owned committed partnership a cell phone. and felt that their spouse or partner was distracted by his or her cell phone when they were together, Statistic 25% of cell phone owners felt that their spouse or partner was distracted by her or his cell phone when they were together. ❖ An important difference between statistics and parameters i that statistics are knowable. In Example 1 above, it is easier for readers to find the sample and statistic rather than the population and the parameter. Unknown Known Population All cell phone owners in a committed relationship Sample A small number of cell phone owners in a committed relationship Parameter Percentage of all cell-phone owners in a committed relationship who felt that their partner was distracted when they were together ★ The letter ★ ︿ prepresents the (p-hat) p represents the Statistic Percentage of the sample who felt their partner was distracted when they were together proportion of a population . proportion of a sample . What Could Possibly Go Wrong? The Problem of Bias ❖ A method is biased if it has a tendency to produce an untrue value. ❖ Bias can enter a survey in three ways: 1. The first it through sampling bias , which results from taking a sample that is “not representative” of the population. ■ “Not representative” means that the sample is fundamentally different from the population. 2. The second way is measurement bias , which comes from asking questions that do not produce a true answer. ■ Example : if we ask people their income, they are likely to inflate the value. In this case, we will get a positive (or “upward”) bias: Our estimate will tend to be too high. ■ Measurement bias occurs when measurements tend to record values larger (or smaller) than the true value. 3. The third way occurs because statistics are naturally biased . ■ Example : if you use the statistic 10x to estimate the mean, you’ll typically get estimates that are ten times too big. Therefore, even when no measuring or sampling bias is present, you must also take care to use an estimator that is not biased. Sampling Bias ❖ Response bias - people tend to respond to such surveys only if they have strong feelings about the results. ➢ Example: Internet polls display a large component of response bias. ❖ KEYPOINT : When reading about a survey, it is important to know: 1. What percentage of people who were asked to participate actually did so 2. Whether the researchers chose people to participate in the survey or people themselves chose to participate. If a large percentage of those chose to participate refused to answer questions, or if people themselves chose whether to participate, the conclusions of a survey are suspect. ❖ Convenience sample Simple Random Sampling Saves the Day ➢ Example: standing at a street corner and stopping whomever you want to ask them to participate in your survey. ❖ Simple Random Sampling (SRS) : we draw subjects from the population at random and without replacement. (once the subject is drawn, it doesn’t get replaced back in the sample). ❖ Without replacement : once a subject is selected for a sample, that subject cannot be selected again. (once the subject is drawn, it cannot be chosen again). ➢ Example: dealing cards from a deck. Once a card is dealt for a hand, no one can get that same card. Example 2: Taking a Simple Random Sample Alberto, Justin, Michael, Audrey, Brandy, and Nicole are in class. Question: Select an SRS of three names from these six names. Solution: First assign each person a number, as shown: Alberto 1 Justin 2 Michael 3 Audrey 4 Brandy 5 Nicole 6 ❖ Nonresponse bias : people who you sent the survey to don’t respond back to you/refuse to part-take in your survey 7.2 KEYPOINT : Statisticians evaluate the method used for a survey, not the outcome of a single survey. Using Simulations to Understand the Behavior of Estimators The three simulations that follow will help measure how well the sample proportion works as an estimator of the population proportion: 1. The first simulation: Imagine doing a survey of 4 people in a very small population out of 8 people. You’ll see that the estimator of the population proportion is accurate (no bias) because of the small sample size. 2. The second simulation: the first simulation is repeated, using a larger population and sample. The estimator is still unbiased, and you will see a perhaps surprising change in precision. 3. The third simulation: will be revealed using a much larger sample size by making the results even more precise. ❖ Sample proportion - comes from the sample, not the population. KEYPOINT : No matter how much different samples we take, the value of (theppopulation proportion) is always the same, but the value ofp ︿ changes from sample to sample. ︿ ❖ Sampling distribution : this term reminds us that p is not just any random outcome; it is a statistic we use to estimate a population parameter. ➢ The bias of a sampling distribution is measured by computing the distance between the center of the sampling distribution and the population parameter. ❖ The estimator has no bias and that on average, it is the same as the parameter value. ❖ Bias : measured as the distance between the mean value of the estimator (the center of the sampling distribution) and the population parameter. ❖ Precision : reflected in the spread of the sampling distribution and is measured by using the standard deviation of the sampling distribution. ➢ Precision, we know that our estimator is, on average, the same as the parameter, but the sampling distribution tells us how far away, typically, the estimator might stray from average. ❖ Standard error (SE): the standard deviation of a sampling distribution ➢ The standard error measures how much our estimator typically varies from sample to sample. ➢ The standard error of the sampling distribution is to measure the precision KEYPOINT : Bias is measured using the center of the sampling distribution: It is the distance between the center and the population value. Precision is measured using the standard deviation of the sampling distribution, which is called the standard error . When the standard error is small, we say the estimator is precise . Sampling Distribution - It is special name for the probability distribution of a statistic. - It gives us probabilities for a statistic. - HOW? It tells us how often we can expect to see particular values of our estimator, and it also gives us important characteristics or the estimator, such as bias and precision. - HOW IS IT USED? It is used for making inferences about a population. KEYPOINT : The precision of an estimator does not depend on the size of the population: it depends only on the sample size. An estimator based on a sample size of 10 is just as precise in a population of 1000 people as in a population of a million. KEYPOINT : Surveys based on a larger sample sizes have smaller standard error (SE) and therefore better precision. Increasing the sample size improves precision. ︿ Finding the Bias and the Standard Error ❖ The bias of p is 0, and the standard error is: Formula 7.1a: √ SE p(1 − p) = n ❖ If the following two conditions are met: 1. Condition 1: The sample must be randomly selected from the population of interest, either with or without replacement. The population parameter to be estimated is the proportion of people (or objects with some characteristic. This proportion is denoted as p. 2. If the sampling is without replacement, the population needs to be much larger that the sample size: at least 10 times bigger is a good rule of thumb. Formula 7.1b: ❖ SE est SE √ est ︿ ︿ p (1 − p) = n is the estimated standard error. Running a Survey 1. Establishing the target population 2. Obtain a list of the target population 3. Select a sample from the list a. Sampled units should be representative of the target population. How do we ensure this? i. Example: a survey of people at Montclair State University who they might vote for in an upcoming election b. Answer : Select a subset of units from the population at random. i. Unreliable methods of selecting samples (How notto select a sample): 1. Convenience sample : a. Pick units that are easy to measure i. In other words, a surveyor may pick the people who are most convenient to them; example : going to the Student Center to convey a survey yet this wouldn’t work because it doesn’t represent MSU as a whole. 2. Judgement sampling : a. Pink units you judge as representative i. A surveyor goes out and uses their judgement to pick out a “typical” person from the population; example : everyone has inherent biases and might pick people who seem friendly or are like you yet these “typical” people don’t represent the entire population. 3. Voluntary response sampling : a. Pick units who respond voluntarily i. People have to volunteer to respond surveys, mostly because they have strong feelings about the survey; example: online surveys/polls c. When random sample is large enough, it should have characteristics that mirror the characteristics of the population. 4. Obtain data from the sampled units ★ The quality of a survey depends on the method used of the survey. ❖ Sampling Error : any difference that exists between a statistic and its corresponding parameter.