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Unformatted text preview: M316 Chapter 17 Dr. Berg Part II Review Random sampling and randomized comparative experiments are perhaps the most important statistical inventions of the 20th century. Both were slow to catch on, but are essential to the reliability of the results. Statistical inference draws conclusions about a population on the basis of sample data and uses probability to indicate how reliable the conclusions are. A confidence interval estimates an unknown parameter. A significance test shows how strong the evidence is for some claim about a parameter. Part II Summary Here are some important skills. A. SAMPLING 1 Identify the population in a sampling situation. 2 Recognize bias due to voluntary response samples and other inferior sampling methods. 3 Use software or Table B of random digits to select a simple random sample (SRS) from a population. 4 Recognize the presence of under‐coverage and non‐response as sources of error in a sample survey. Recognize the effect of the wording of questions on the responses. 5 Use random digits to select a stratified random sample from a population when the strata are identified. B. EXPERIMENTS 1 Recognize whether a study is an observational study or an experiment. 2 Recognize bias due to confounding of explanatory variables with lurking variables in either an observational study or an experiment. 3 Identify the factors (explanatory variables), treatments, response variables, and individuals or subjects in an experiment. 4 Outline the design of a completely randomized experiment using a diagram that shows the sizes of the groups, the specific treatments, and the response variable. 5 Use software or Table B of random digits to carry out the random assignment of subjects to groups in a completely randomized experiment. 6 Recognize the placebo effect. Recognize when the double‐blind technique should be used. 7 Explain why randomized comparative experiments can give good evidence for cause‐and‐effect relationships. 1 M316 Chapter 17 Dr. Berg € C PROBABILITY 1 Recognize that some phenomena are random. Probability describes the long‐ run regularity of random phenomena. 2 Understand that the probability of an event is the proportion of times the event occurs in very many repetitions of a random phenomenon. Use the idea of probability as long‐run proportion to think about probability. 3 Use basic probability rules to detect illegitimate assignments of probability: Any probability must be a number between 0 and 1, and the total probability assigned to all possible outcomes must be 1. 4 Use basic probability rules to find the probabilities of events that are formed from other events. The probability that an event does not occur is 1 minus its probability. If two events are disjoint, the probability that one or the other occurs is the sum of their individual probabilities. 5 Find probabilities in a discrete probability model by adding the probabilities of the outcomes. Find probabilities in a continuous probability model as areas under a density curve. 6 Use the notation of random variables to make compact statements about random outcomes, such as P ( x ≤ 4 ) = 0.3. Be able to interpret such statements. D. SAMPLING DISTRIBUTIONS 1 Identify parameters and statistics in a statistical study. € 2 Recognize the fact of sampling variability: a statistic will take different values when you repeat a sample or experiment. 3 Interpret a sampling distribution as describing the values taken by a statistic in all possible repetitions of a sample or experiment under the same conditions. 4 Interpret the sampling distribution of a statistic as describing the probabilities of its possible values. E. THE SAMPLING DISTRIBUTION OF A SAMPLE MEAN 1 Recognize when a problem involves the mean x of a sample. Understand that x estimates the mean µ of the population from which the sample is drawn. 2 Use the law of large numbers to describe the behavior of x as the size of the sample increases. € 3 Find the mean and standard deviation of a sample mean x from an SRS of size n when the mean µ and standard deviation σ of the population are known. € 4 Understand that x is an unbiased estimator of µ and that the variability of x about its mean µ gets smaller as the sample size increases. € 5 Understand that x has approximately a Normal distribution when the sample is large (central limit theorem). Use this Normal distribution to calculate € € probabilities that concern x . € F. GENERAL RULES OF PROBABILITY 1 Use Venn diagrams to picture relationships among several events. € 2 Use the general addition rule to find probabilities that involve overlapping events. 2 M316 Chapter 17 Dr. Berg 3 Understand the idea of independence. Judge when it is reasonable to assume independence as part of a probability model. 4 Use the multiplication rule for independent events to find the probability that several independent events occur. 5 Use the multiplication rule for independent events in combination with other probability rules to find the probabilities of complex events. 6 Understand the idea of conditional probability. Find the conditional probabilities for individuals chosen at random from a table of counts of possible outcomes. 7 Use the general multiplication rule to find P ( A and B) from P(A) and P(AB). 8 Use tree diagrams to organize multi‐stage conditional probability problems. G BINOMIAL DISTRIBUTIONS € 1 Recognize the binomial setting: a fixed number n of independent success‐ failure trials with the same probability p of success on each trial. 2 Recognize and use the binomial distribution of the count of successes in a binomial setting. 3 Use the binomial probability formula to find probabilities of events involving the count X of successes in a binomial setting for small values of n. 4 Find the mean and standard deviation of a binomial count X. 5 Recognize when you can use the Normal approximation to a binomial distribution. Use the Normal approximation to calculate probabilities that concern a binomial count X. H. CONFIDENCE INTERVALS 1 State in nontechnical language what is meant by “95% confidence” or other statements of confidence in statistical reports. 2 Know the four‐step process (p350) for any confidence interval. 3 Calculate a confidence interval for the mean µ of a Normal population with known standard deviation σ, using the formula x ± z * σ / n . 4 Understand how the margin of error of a confidence interval changes with the sample size and the level of confidence C. 5 Find the sample size required to obtain a confidence interval of a specified € margin of error m when the confidence level and other information are given. 6 Identify sources of error in a study that are not included in the margin of error of a confidence interval, such as under‐coverage or non‐response. I. SIGNIFICANCE TESTS 1 State the null and alternative hypotheses in a testing situation when the parameter in question is a population mean µ. 2 Explain in nontechnical language the meaning of the P‐value when you are given the numerical value of P for a test. 3 Know the four‐step process (p372) for any significance test. 4 Calculate the one‐sample z test statistic and the P‐value for both one‐sided and two‐sided tests about the mean µ of a population. 3 M316 Chapter 17 Dr. Berg 5 Assess the statistical significance at standard levels α, either by comparing P with α or by comparing z with standard Normal critical values. 6 Recognize that significance testing does not measure the size or importance of an effect. Explain why a small effect can be significant in a large sample and a large effect can fail to be significant in a small survey. 7 Recognize that any inference procedure acts as if the data were properly produced. The z confidence interval and test require that the data be and SRS from the population. Exercises 17.3 Tom Clancy’s Writing Different types of writing can sometimes be distinguished by the lengths of the words used. A student interested in this fact wants to study the lengths of words used by Tom Clancy in his novels. She opens a Clancy novel at random and records the lengths of each of the first 250 words on the page. What is the population in this study? What is the sample? What is the variable measured? 17.6 Support Groups for Breast Cancer Does participation in a support group extend the lives of women with breast cancer? There is no good evidence for this claim, but it was hard to carry out randomized comparative experiments because breast cancer patients believe that support groups help and want to be in one. When the first such experiment was finally completed, it showed that support groups have no effect on survival time. The experiment assigned 235 women with advanced breast cancer to two groups: 158 to “expressive group therapy” and 77 to a control group. a) Outline the design of this experiment. b) Use Table B at line 110 to choose the first 5 members of the control group. 17.10 Estimating Blood Cholesterol The distribution of blood cholesterol level in the population of young men aged 20 to 34 years is close to Normal with standard deviation σ = 41 milligrams per deciliter (mg/dl). You measure the blood cholesterol of 14 cross‐country runners. The mean level is x = 172 mg/dl. Assuming that σ is the same as in the general population, give a 90% confidence interval for the mean level µ among € cross‐country runners. € 17.11 Testing Blood Cholesterol The mean blood cholesterol level for all men 20 to 34 years of age is µ = 188 mg/dl. We suspect that the mean for cross‐country runners is lower. State the hypotheses, use the information in 17.10 to find the test statistic, and give the P‐ value. Check it against significance levels of α = 0.10 , α = 0.05 , and α = 0.01. € € 4 € € M316 Chapter 17 Dr. Berg 17.12 Smaller Margin of Error How large a sample is needed in 17.10 to cut the margin of error to 5 mg/dl? 17.23 Moving Up A study of social mobility in England looked at the social class reached by the sons of lower‐class fathers. Social classes are numbered from 1 (low) to 5 (high). Take the random variable X to be the class of a randomly chosen son of a father in Class 1. The study found the distribution of X is Son’s Class 1 2 3 4 5 Probability 0.48 0.38 0.08 0.05 0.01 a) Check that this satisfies the properties of a discrete probability model. b) What is P(X≤3)? c) What is P(X<3)? 17.24 The Addition Rule The addition rule for probabilities is not always true. Give an example of real‐ world events for which this rule in not true. 5 ...
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