Probability & Statistics Chapter 10 Random Samples Probability & Statistics Chapter 10

Simple Random Samples A simple random sample of n measurements from a population is a subset of the population selected in such a manner than every sample of size n from the population has an equal chance of being selected. Additionally, each individual in the population has an equal chance of being selected for the sample.

Discussion Players of the Colorado Lottery each pay $1.00 and select any six different numbers from the group of numbers 1 through 42. If a player’s group of six numbers matches the winning group of six numbers (selected by simple random sampling), then that player is a winner of a grand prize of at least $1.5million. Is the number 25 as likely to be selected in the winning group of six numbers as the number 5? Could all the winning numbers be even? Your friend always plays the numbers 1, 2, 3, 4, 5, 6. Could she ever win? Explain your answer. Answer 1: Yes. Because the winning numbers constitute a simple random sample, each number from 1 through 42 has an equal chance of being selected. Answer 2: Yes, since six even numbers is one of the possible groups of six numbers. Each group of six numbers has an equal chance of being selected in a simple random sample. Answer 3: Yes. In a simple random sample, the listed group of six numbers is as likely as any of the 5,245,786 possible groups of six number to be selected as the winner.

How do we get random samples? One way to pick a random sample is by placing each individual’s name or number on a card in a hat. Then, mix up the cards and select the number of individuals needed for your sample. A simpler way to collect a random sample is by using a random number table or a random number generator. Many computers and calculators have internal programs allowing them to become random number generators. Let’s look at how to use a random number table.

Random Number Tables Below is an example of two rows from a random number table. 19281 59640 15221 46079 09961 05371 53362 34101 29553 83834 51350 65472 Example: Suppose you need to know if the emission systems of the latest shipment of Toyotas satisfy pollution-control standards. You want to pick a random sample of 10 cars from this shipment of 500 cars and test them. Let’s use the two rows of random numbers (above) to help us select our sample.

Random Number Tables 19281 59640 15221 46079 09961 05371 53362 34101 29553 83834 51350 65472 Since there are 500 cars altogether, each car will be assigned a number 1 – 500. Since the largest number (500) has three digits, we will need to use our random numbers in sets of three. 192 815 964 015 221 460 790 996 105 371 533 623 410 129 553 838 345 135 065 472 Now, read through the numbers above (starting with 192), and find 10 numbers (500 or less) that can be used for our sample. The cars selected are: 192, 15, 221, 460, 105, 371, 410, 129, 345, 135.

Simulations Another important use of random number tables is in simulation. A simulation is a numerical facsimile or representation of a real-world phenomenon. We use the word simulation to refer to the process of providing numerical imitations of “real” phenomena.

Sampling Techniques Random Sampling: Use a simple random sample from the entire population. Stratified Sampling: Divide the entire population into distinct subgroups called strata. The strata are based on a specific characteristic such as age, income, education level, and so on. All members of a stratum share the specific characteristic. Draw random samples from each stratum. Systematic Sampling: Number all members of the population sequentially. Then, from a starting point selected at random, include every nth member of the population in the sample. Cluster Sampling: Divide the entire population into pre-existing segments or clusters. The clusters are often geographic. Make a random selection of clusters. Include every member of each selected cluster in the sample. Multistage Sampling: Use a variety of sampling methods to create successively smaller groups at each stage. The final sample consists of clusters. Convenience Sampling: Create a sample by using data from populations members that are readily available. Please note that convenience sampling is statistically dangerous and often yields untrustworthy results.

More Terms to Know A sampling frame is a list of individuals from which a sample is actually selected. Undercoverage results when population members are omitted from the sample frame. A sampling error is the difference between measurements from a sample and corresponding measurements from the respective population. It is caused by the fact that the sample does not perfectly represent the population. A nonsampling error is the result of poor sample design, sloppy data collection, faulty measuring instruments, bias in questionnaires, and so on. Bias - sampling methods that by their nature tend to over- or underemphasize some characteristics of the population