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AP Statistics Review Part V & Part VI & Part VII: From the Data at Hand to the World at Large Learning About the World Inference When Variables Are Related.

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Presentation on theme: "AP Statistics Review Part V & Part VI & Part VII: From the Data at Hand to the World at Large Learning About the World Inference When Variables Are Related."— Presentation transcript:

1 AP Statistics Review Part V & Part VI & Part VII: From the Data at Hand to the World at Large Learning About the World Inference When Variables Are Related

2 FROM THE DATA AT HAND TO THE WORLD AT LARGE Part V: Chapters 18 - 22

3 SAMPLING DISTRIBUTION MODELS

4 Sampling Distributions A sampling distribution is the distribution of a statistic, such as x-bar or p-hat, from all possible samples of a given size (n) from a population of size N. A certain proportion (p) of the population may display a particular characteristic. In each randomly selected sample of sze n, that same characteristic will be seen in a proportion (p- hat) of the sample. We use p-hat (our sample proportion) as an estimate of p (the population proportion). The same is true for the sample mean (x-bar) as an estimate of the population mean ( μ ).

5 Modeling the Distribution of Sample Proportions The sampling distribution of sample proportions has these characteristics. Shape – The distribution of sample proportions is approximately normally distributed provided that np ≥ 10 (there are at least 10 successes) and n(1-p) ≥ 10 (there are at least 10 failures) Center – The mean of the sampling distribution of sample proportions, (p-hat) is centered at the mean of the population, p, regardless of sample size. (We say that p-hat is an unbiased estimator of p). Spread – The standard deviation of sampling distribution of sample proportions is given by the formula, provided that 10n ≥ N (since we are sampling without replacement, the sample size should be less than 10% of the population size in order to use this formula for standard error).

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7 Modeling the Distribution of Sample Means For categorical variables, we are interested in a proportion (p). For quantitative variables, a parameter of interest is the average or mean ( μ ). The sampling distribution of sample means has these characteristics. Shape – As sample size increases, the distribution of sample means tends toward an approximately normal distribution (Central Limit Theorem). Center – The mean of the sampling distribution of sample means,, is centered at the mean of the population, μ, regardless of sample size (We say that x-bar is an unbiased estimator of μ ). Spread – The standard deviation of the sampling distribution of sample means is given by the formula.

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9 Central Limit Theorem The Central Limit Theorem (CLT) says that means of repeated samples will tend to follow a normal model if the sample size is “large enough”; this true no matter what shape the population distribution has! If the population is not Normal, the sample size should be at least 30 for the CLT to apply. Avoid confusing the distribution of a sample with the sampling distribution model for sample means. The distribution of a (one) sample will, as n increases, look more like the population from which it was taken.

10 Standard Error For both sampling distribution models (proportions and means), we have assumed that we know the population parameters ρ or μ and σ. In most cases, it is not possible to know these values. Consequently, we must use what we do know and estimate the standard deviations of sampling distributions. When this happens, we call the estimated standard deviation the standard error.

11 Standard Error (cont.) For a proportion, the standard error of p-hat is: For a sample mean, the standard error of x-bar is: We estimate a population parameter, p, by using a sample statistic,. For this reason, is called a point estimate for p. In general, if the expected value of the estimator equals the corresponding parameter, the statistic is called an unbiased estimator. Since, is an unbiased estimator for p. (Similarly, is a point estimate for μ, and since, is an unbiased estimator for μ.)

12 Standard Error (cont.) We have seen that sample statistics vary. So rather than simply giving a single value for our estimate of the population proportion, we use our sample estimate,, to build a range of plausible values. To do this, we want to establish an interval that we believe (at least with some degree of certainty) contains the true value of our population parameter. This interval forms an interval estimate for the parameter, p, and is called a confidence interval. Because of sampling variability, we can never say we are 100% certain; therefore, the confidence level we choose indicates the degree of certainty we have that our interval captures the true value of the parameter.

13 CONFIDENCE INTERVALS FOR PROPORTIONS

14 Creating and Interpreting a Confidence Interval What We Say If we choose 95% as our confidence level, we build an interval that stretches about 2 SE’s from in either direction. Then we say: We are 95% confident that the true proportion is between (calculated lower limit) and (calculated upper limit). This is a correct interpretation of the confidence interval.

15 Creating and Interpreting a Confidence Interval It is also possible to interpret the confidence level. A correct interpretation of the confidence level would be: 95% confidence means that, in the long run, 95 out of 100 intervals calculated using the same procedure would (using random samples of the same size from the same population) capture the true population proportion. Commonly used values for confidence levels are 90%, 95%, and 99%, but other levels may be chosen.

16 Creating and Interpreting a Confidence Interval Know the difference between confidence interval and confidence level and be sure to give the correct interpretation. It’s easy to get interval and level confused.

17 Creating and Interpreting a Confidence Interval The general form of all confidence intervals is: Estimate ± (Critical Value)(Standard Error) The desired confidence level determines the critical value (the number of standard deviations on either side of the estimate). The critical value (CV) is denoted by z*. The most commonly used values are: Level z* 90%1.645 95%1.960 99%2.576

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19 Conditions for a One-Proportion z-Interval Randomization Condition – proper randomization techniques used to collect the data. 10% Condition – sample size is less than 10% of the population. Success/Failure Condition – have at least 10 “successes” and 10 “failures”.

20 Margin of Error The distance of the endpoints of the interval from is called the margin of error (ME), in this case. The margin of error is tied to the level of confidence. (Surveys in the newspaper or on TV usually report a margin of error based on a 95% confidence level.) The more confident you want to be, the larger the margin of error will be. More confidence implies a wider interval. Since the standard error of is tied to the sample size, as the sample size increases, the value of the standard error deceases and results in a narrower interval.

21 Things to Remember About a Parameter A parameter does not vary – there is only One true value of a parameter. You DO NOT KNOW the ture value of the parameter. If you did, you would not need a confidence interval. NEVER use when referring to the value of the parameter, p.

22 Things to Remember About Interpreting a Confidence Interval A confidence interval makes a statement about the true population proportion, and your interpretation of the interval should be about the population proportion. Keep your interpretation to what you know – do not claim to know too much. A sentence or two should be sufficient. Sample statistics vary with each sample. Not every interval you compute will capture the true population proportion. Your interpretation of the confidence interval should reflect how confident you are that your interval has managed to capture the true value; your confidence is in the method used to produce the interval.

23 TESTING HYPOTHESES ABOUT PROPORTIONS

24 Confidence Intervals and Hypothesis Tests Confidence intervals, just one form of statistical inference, allow us to use a sample statistic to make a statement about how confident we are that a population parameter lies within certain limits. Another form of statistical inference is a hypothesis test or test of significance. We use this form of statistical inference when a particular conjecture or claim (hypothesis) has been made about the true value of a population parameter. Again, we use sample statistics to help us decide whether or not to believe the hypothesis. We want to know if there is enough evidence to support (NOT prove) the hypothesis or to reject it.

25 The Reasoning of Hypothesis Testing Hypothesis The null hypothesis: The term null comes from the idea that there is “no difference” between the hypothesized value and the true population parameter. H 0 : p=p 0 (hypothesized value) The null hypothesis MUST have a sign of equality and MUST be written using population parameters and NEVER sample statistics. You MUST use standard notation (don’t make up your own symbols). You should write your hypothesis in both symbols and words. Your final decision MUST always be stated in terms of rejecting or failing to reject the null hypothesis.

26 The Reasoning of Hypothesis Testing (cont.) The alternative hypothesis: the alternative hypothesis is what would be accepted as true if the null hypothesis were to be rejected. It can take one of three different forms; For proportions, a one-sided alternative would be either H a : p p 0. A two-sided alternative would be H a : p ≠ p 0. The form used will depend on the study being done and should be clear from the context of the problem.

27 The Reasoning of Hypothesis Testing (cont.) Plan Collect data that can be used to calculate a sample statistic to test against the null hypothesis. Decision Rule: Decide what will be “convincing evidence” to reject the null hypothesis. Mechanics Check the conditions for the sampling distribution model. Calculate a test statistic. A test statistic is the standardized value of the sample statistic with respect to the hypothesized value.

28 The Reasoning of Hypothesis Testing (cont.) In general, it has the form For a proportion, the test statistic is given by Note: When computing SD for the test statistic, we use p 0, because our model is based on the conjecture or claim (null hypothesis). When computing the SE for a confidence interval, we use p-hat because we do not have a hypothesized parameter.

29 The Reasoning of Hypothesis Testing (cont.) Calculate the p-value. The p-value is the probability that a value at least as extreme as the observed value could occur if, in fact, the hypothesized value were true. Use the normal table to convert your test statistic to a p-value. The smaller the p-value, the less likely it is that our sample came from a population with the parameter value stated in our null hypothesis. In other words, the smaller the p-value, the stronger the evidence is against the null hypothesis.

30 The Reasoning of Hypothesis Testing (cont.) Conclusion Make a decision about whether to reject or fail to reject the null hypothesis based upon the p- value. Reject H 0 if the calculated p-value is less than the chosen α –level. We say that the result is statistically significant. Your statement MUST include Decision (reject H 0 in favor of H a or fail to reject H 0 ). Criteria for decision (p-value compared with α –level or test statistic compared with critical value). Context (use the words of the problem to restate your decision).

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32 MORE ABOUT TESTS

33 Possible Errors When Hypothesis Testing Because of the variability of sample data, we may draw a wrong conclusion despite having followed correct procedures. Whenever we make decisions based on sample data, there is a risk of error. These errors are of two types, Type I and Type II.

34 Type I Error A Type I Error is the consequence of rejecting a null hypothesis that is in fact true. The probability of making a Type I Error is equal to the value of α, the level of significance.

35 Type II Error A Type II Error is the consequence of failing to reject a null hypothesis that is in fact false. The probability of this type of error is called β. The value of β changes depending on the value of the parameter that is chosen as the alternative. Note: Knowing how to calculate β is NOT a requirement of the AP curriculum. You ARE expected to understand the difference between α and β and how they affect each other.

36 Power The probability that a test will correctly reject a null hypothesis that is in fact false is called the power of the test. POWER = 1 - β. Power is the complement of a Type II Error. High power is desired because it indicates the sensitivity of the test to specific values of the alternative hypothesis.

37 Ways to Reduce Error Type I Error ( α ) – decrease the α -level. Type II Error ( β ) – increase the α -level. Type I and Type II Errors – increase the sample size, n. Power (1 - β ) – increase the sample size, n, or decrease β, or select a different, more extreme alternative.

38 COMPARING TWO PROPORTIONS

39 The Sampling Distribution for

40 Conditions for the Difference of Two Proportions Independence Condition – the two sample groups are independent of one another. Randomization 10% Condition Success/Failure Condition

41 Two-Proportion z-Interval

42 Two Proportion z-Test The procedure for this test is essentially the same as for a one-proportion z-test with one slight difference: the null hypothesis is that there is no difference between the two proportions. H 0 : p 1 =p 2 (H 0 : p 1 -p 2 =0) The alternative hypothesis is determined by the kind of difference you expect to get in the particular situation ( ).

43 Two Proportion z-Test The standard error in this case is pooled (combine the two proportions). It is okay to pool the proportions because the null hypothesis assumes that p 1 =p 2 and thus the two populations are identical for the attribute being studied.

44 Two Proportion z-Test It is highly unlikely that you will have to do this calculation by hand. The calculator two-proportion z-test calculates the z-score automatically and uses the correct standard error.

45 What You Need to Know Identify parameters and statistics in a sample or experiment. Recognize the fact of sampling variability: a statistic will take different values when you repeat a sample or experiment. 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. Describe the bias and variability of a statistic in terms of the mean and spread of its sampling distribution. Understand that the variability of a statistic is controlled by the size of the sample. Statistics from larger samples are less variable.

46 What You Need to Know

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50 PRACTICE PROBLEMS

51 #1 Suppose that 35% of all business executives are willing to switch companies if offered a higher salary. If a headhunter randomly contacts an SRS of 100 executives, what is the probability that over 40% will be willing to switch companies if offered a higher salary? a).1469 b).1977 c).4207 d).8023 e).8531

52 #2 The average number of daily emergency room admissions at a hospital is 85 with a standard deviation fo 37. In an SRS of 30 days, what is the probability that the mean number of daily emergency admissions is between 75 and 95? a).1388 b).2128 c).4090 d).5910 e).9474

53 #3 A confidence interval estimate is determined from a SRS of n students. Which of the following will result in a smaller margin of error? I. A smaller confidence level II. A smaller sample size a) I only b) II only c) both I and II

54 #4 A survey was conducted to determine the percentage of high school students who planned to go to college. The results were stated as 82% with a margin of error of 5%. What is meant by +/- 5%? a) Five percent of the population were not surveyed. b) In the sample, the percentage of students who plan to go to college was between 77% and 87% c) The percentage of the entire population of students who plan to go to college is between 77% and 87% d) It is unlikely that the given sample proportion result would be obtained unless the true percentage was between 77% and 87% e) Between 77% and 87% of the population were surveyed.

55 #5 A USA Today “Lifeline” column reported that in a survey of 500 people, 39% said they watch their bread while it’s being toasted. Establish a 90% confidence interval estimate for the percentage of people who watch their bread being toasted. a) 39% +/-.078% b) 39% +/- 2.2% c) 39% +/- 2.8% d) 39% +/- 3.6% e) 39% +/- 4.3%

56 #6 A politician wants to know what percentage of the voters support her position on a hot issue. What size voter sample should be obtained to determine with 90% confidence the support level to within 4%? a) 21 b) 25 c) 423 d) 600 e) 1691

57 #7 Which of the following are true statements? I. Hypothesis tests are designed to measure the strength of the evidence against the null hypothesis. II. A well-planned test should result in a statement either that the null hypothesis is true or that it is false. III. The alternate hypothesis is one-sided if there is interest in deviations from the null hypothesis in only one direction. a) I and II b) I and III c) II and III d) I, II, and III e) None of the above

58 #8 A building inspector believes that the percentage of new construction with serious code violations may be even greater than the previously claimed 7%. She conducted a hypothesis test on 200 new homes and finds 23 with serious code violations. Is this strong evidence against the 7% claim? a) Yes, because the P-value is.0062 b) Yes, because the P-value is 2.5 c) No, because the P-value is only.0062 d) No, because the P-value is over 2 e) No, because the P-value is.045

59 #9 In a survey of 9700 T.V. viewers, 40% said they watch network news programs. Find the margin of error for this survey if we want 95% confidence in our estimate of the percent of T.V. viewers who watch network news programs. a) 1.12% b) 1.28% c) 0.731% d) 0.975%

60 #10 Which is true about a 99% confidence interval based on a given sample? I. The interval contains 99% of the population. II. Results from approximately 99% of all samples will capture the true parameter in their respective intervals. III. The interval is wider than a 95% confidence interval would be. a) I only b) II only c) III only d) II and III only e) None

61 #11 To maintain customer satisfaction, an online catalog company wants to have on-time delivery for 90% or better of the orders they ship. The company has been shipping their orders via UPS and FedEx but will switch to a new, cheaper delivery service called ShipFast unless there is evidence that this service cannot meet the delivery goal of 90% or better. As a test the company sends a random sample of orders via ShipFast, and then makes follow-up phone calls to see if these orders arrived on time. Which hypotheses should they test? a)H 0 : p<0.90 H a : p=0.90 b)H 0 : p>0.90 H a : p=0.90 c)H 0 : p=0.90 H a : p<0.90 d)H 0 : p=0.90 H a : p ≠ 0.90 e)H 0 : p=0.90 H a : p>0.90

62 #12 A researcher investigating whether runners are less likely to get colds than non-runners found a P-value of 3%. This means that: a) 3% of runners get colds. b) 3% fewer runners get colds. c) There’s a 3% chance that runners get fewer colds. d) There's a 3% chance our assumption of no difference in number of colds whether a runner or not is incorrect. e) There’s a 3% chance that the sample statistic or more extreme will occur assuming there is no difference between number of colds whether a runner or not.

63 LEARNING ABOUT THE WORLD Part Vi: Chapters 23 - 25

64 INFERENCE ABOUT MEANS

65 The t-Distributions: A Sampling Distribution for Means By the Central Limit Theorem (CLT), the means of repeated samples tend to follow a normal model as long as the population distribution is normal or the sample size is large enough. According to the CLT, the formula for the standard deviation of the sampling distribution for means is When we have to estimate with, the standard error of the sampling distribution for means is Because σ is unknown, the standard error is used in place of the standard deviation, and the shape of the sampling distribution changes. The new model is a family of distributions called the t-distribution.

66 The t-Distributions The shape of the t-distributions (unimodal, symmetric, and bell-shaped) is connected to the sample size by a parameter called the degrees of freedom. The increased variation due to small sample size increases the probability in the tails. The degrees of freedom (df) determine the particular t-distribution. Degrees of freedom (df) equals (n – 1).

67 Sampling Distribution for Means

68 Conditions for Using t-Distributions Randomization Condition 10% Condition Nearly Normal Condition – the data come from a unimodal, symmetric, bell-shaped distribution. This can be verified by constructing a histogram or a normal probability plot of the sample data. If the population is approximately normal, the t- statistic is appropriate regardless of sample size. For n ≤ 40, the t-statistic is appropriate provided there are no outliers. For 15 ≤ n<40, the t-statistic is appropriate provided there are no outliers or strong skewness. For n<15, the t-statistic is appropriate provided the distribution of the sample data is approximately normal.

69 One-Sample Confidence Interval for the Mean ( σ unknown) Because we are using s to estimate, we use a t-critical value for confidence intervals and a t-test statistic for hypothesis tests.

70 One-Sample t-Confidence Interval

71 Hypothesis Test for Means The format for all hypothesis tests is essentially the same as it was for proportions.

72 Determining Sample Size Once you have decided on a confidence level and margin of error, you calculate the sample size from the appropriate confidence interval formula. You should always round the sample size (n) UP to the nearest integer. Sample size should be a whole number

73 INFERENCE FOR TWO INDEPENDENT SAMPLES

74 Comparing Two Means As is the case of proportions, the parameter of interest is the difference between two population means, μ 1 - μ 2. The mechanics are the same.

75 The Sampling Distribution for Two Independent Means The mean of the sampling distribution of is μ 1 - μ 2. The standard error, because we substitute s for σ, of the sampling distribution is The assumptions and the conditions for inference for two-sample means are the same as those for one-sample means, with the added condition that the samples are independent.

76 A Two-Sample t-Interval for Means The formula for calculating the actual degrees of freedom is rather complicated, and generally is not a whole number (the TI calculator gives this number).

77 A Two-Sample t-Interval for Means When using the TI calculator, you are asked to choose “Pooled: No Yes.” Choose NO. The word pooled here does not have the same meaning as it did for two proportions.

78 Test for the Difference Between Two Means When you record a p-value, be certain that it is a number from 0 to 1. Do not carelessly copy the calculator output! Probabilities cannot exceed 1!

79 A SPECIAL CASE: TWO SAMPLES OR ONE?

80 Matched Pairs What should we do when the two samples are not independent. What if the data are for the same people at two different times. It would not make sense to get averages before and after, and do a two-sample t- test for means. Besides it would violate the condition of independent samples. We really want to know the amount of change for each individual. So we subtract first and create a single set of data. Now we use a one-sample procedure.

81 Matched Pairs There are many situations that call for data to be treated in this way, but we must have a good reason for pairing the data. Just because two samples have the same number of data values is not a reason for pairing them. It is up to you to recognize when a matched pairs procedure is called for. For matched pairs, we use the same procedures we used for the one-sample t- interval or the one-sample t-test for the mean.

82 What You Need to Know Why the t-distribution is used to model the sampling distribution of sample means? Properties of the t-distribution, the sampling distribution for means. What degrees of freedom is and how to calculate. Conditions for using t-distribution. Calculate one-sample t-confidence interval for means. Conduct one-sample t-hypothesis test for means.

83 What You Need to Know Determine sample size for means. Sampling distribution for two means and conditions required for inference. Calculate two-sample t-confidence interval for means. Conduct two-sample t-hypothesis test for the difference of means. Recognize a matched pairs situation. Calculate matched pairs t-confidence interval and t-hypothesis test.

84 PRACTICE PROBLEMS

85 #1 In preparing to use a t procedure, suppose we were not sure if the population was normal. In which of the following circumstances would we not be safe using a t procedure? a)A stemplot of the data is roughly bell shaped. b)A histogram of the data shows moderate skewness. c)A stemplot of the data has a large outlier. d)The sample standard deviation is large.

86 #2 The weight of 9 men have mean of 175 lbs and a standard deviation of 15 lbs. What is the standard error of the mean? a)58.3 b)19.4 c)5 d)1.7 e)None of the above.

87 #3 What is the critical value t* which satisfies the condition that the t distribution with 8 degrees of freedom has probability 0.10 to the right of t*? a)1.397 b)1.282 c)2.89 d)0.90 e)None of the above. The answer is _____.

88 #4 Suppose we have two SRSs from two distinct populations and the sample are independent. We measure the same variable for both samples. Suppose both populations of the values of these variables are normally distributed but the means and the standard deviations are unknown. For purposes of comparing the two means, we use a)Two-sample t procedures b)Matched pairs t procedures c)z procedures d)The least-squares regression line e)None of the above.

89 #5 You want to compute a 90% confidence interval for the mean of a population with unknown population standard deviation. The sample size is 30. The value of t* you would use for this interval is a)1.96 b)1.645 c)1.699 d).90 e)1.311 f)None of the above.

90 #6 A 95% confidence interval for the mean reading achievement score for a population of third-graders is (44.2, 54.2). The margin of error of this interval is a)95% b)5 c)2.5 d)54.2 e)The answer cannot be determined from the information given.

91 #7 Using the same set of data, you compute a 95% confidence interval and a 99% confidence interval. Which of the following statements is correct? a)The intervals have the same width. b)The 99% interval is wider. c)The 95% interval is wider. d)You cannot determine which interval is wider until you know n and s.

92 #8 To use the two-sample t procedure to perform a significance test on the difference between two means, we assume a)The populations’ standard deviations are known. b)The samples from each population are independent. c)The distributions are exactly normal in each population. d)The sample sizes are large. e)All of the above.

93 #9 In a test for acid rain, 49 water samples showed a mean pH level of 4.4 with a standard deviation of 0.35. Find a 90% confidence interval estimate for the mean pH level. a)4.4 ± 0.01175 b)4.4 ± 0.0839 c)4.4 ± 0.315 d)4.4 ± 0.35 e)None of the above. The answer is _____.

94 Part VII: Chapters 26 & 27 Inference When Variables Are Related

95 INFERENCE WHEN VARIABLES ARE RELATED

96 We have already reviewed inference for proportions (z-intervals and tests) and inference for means (t-intervals and tests). In this part of our review, we look at tests for one and two categorical variables. The data for these tests are usually presented in some form of table divided into categories and must be in the form of counts for each category.

97 COMPARING COUNTS

98 The Chi-Square Statistic There are three inference tests involving categorical variables and counts. These tests introduce a new statistic called the chi-squared statistic, which has the same form for all three tests: Where O stands for the observed value and E stands for the expected value.

99 The Chi-Square Distribution The χ 2 distribution is a family of distributions. Each individual distribution is specified by its degrees of freedom. Unlike the t-distribution, the value of n in this case is not the sample size but rather the number of categories. The χ 2 distribution is skewed to the right and has only positive values. Because there are only positive values, all hypothesis tests are right tailed tests and the p-value is computed by calculating the area under the curve to the right of the computed χ 2 test statistic.

100 THE CHI-SQUARE TESTS

101 Goodness-of-Fit Test One categorical variable – one sample. Used to determine how well a set of observed values matches a set of expected values. In other words, how well does a sample distribution match the hypothesized population distribution? The degree of freedom is the number of categories minus 1 (#cat. – 1). The expected values (E=np) are calculated in one of two ways: Values for each category are expected to all have a uniform distribution. Values are expected to follow a distribution specified according to some stated condition.

102 Assumptions/Conditions for Chi-Square Test for Goodness-of-Fit Counted Data Condition: Make sure the data are listed in the form of counts for each category. Precents or proportions WON’T DO – convert them to counts (rounded to whole numbers) if necessary. Randomization Condition: The individual cases should be a random sample from the population of interest. Expected Cell-Frequency Condition: There are at least 5 cases in each expected cell.

103 Chi-Square Test for Goodness-of-Fit Null and alternative hypotheses are usually written in words. The null hypothesis is a statement that the distributions are consistent or the same. The alternative hypothesis is a statemnt that the distributions are not consistent or the same.

104 Chi-Square Test for Goodness-of-Fit

105 If the expected distribution is uniform, the percentage would be 100%/# of cells. For example, we expect digits in a random- number table to appear with equal frequencies. The percentage for each digit would be 10%. We would use these percentages to calculate the expected counts. If you are asked to identify the cell that contributes the most to the χ 2 test statistic, just look at the individual components that contribute to your sum.

106 Test of Independence The “Chi-Square 2 Test of Independence” is used to determine if, in a single population, there is an association between two categorical variables. The data are presented in a two-way table. The null and alternative hypotheses are written in words. The null hypothesis is a statement that the two categories are independent or not related. The alternative hypothesis is that the two categories are depentent or related.

107 Chi-Square Test of Independence The Expected Values are calculated as; The degrees of freedom are calculated as; Although, we reject the null hypothesis of independence, we cannot interpret our small p-value as proof of causation. A failure of independence does not prove a cause-and-effect relationship. There may be lurking variables present.

108 What You Need to Know Be able to write hypotheses for all of the various χ 2 tests. Be able to do the various tests with and without your calculator, decide to reject/retain, and write a conclusion. Know how to find expected values for chi- square tests.

109 PRACTICE PROBLEMS

110 #1 Answer: B

111 #2 A test of independence for data organized in a two-way table relating number of siblings and number of family relocations is conducted using the chi-square distribution. The p-value of the test is.045. If alpha is.05, then which of the following is a valid conclusion of the test? a) The mean is significant. b) We reject the hypothesis that the variables are dependent. c) We accept the hypothesis that the variables are independent. d) We have sufficient evidence to reject the hypothesis that the variables are independent. e) The variables are independent.

112 #3 A chi-square goodness of fit test is used to test whether a 0-9 spinner is “fair” (ie., the outcomes are all equally likely). The spinner is spun 100 times, and the results are recorded. Which member of the chi-square family of curves is used? a)χ 2 (8) b)χ 2 (9) c)χ 2 (10) d)χ 2 (99) e)None of the above

113 #4 A two-way table of counts is analyzed to examine the hypothesis that the row and column classifications are independent. There are 3 rows and 4 columns. The degrees of freedom for the chi-square statistic are a)12 b)11 c)6 d)The minimum of n 1 -1 and n 2 -1 e)None of the above

114 #5 The expected frequency for the cell corresponding to Anger and Males is: a)15.9 b)55.7 c)30.4 d)31.9 e)29.1 In the paper “Color Association of Male and Female 4 th grade School Children”, children were asked to indicate what emotion they associated with the color red.

115 #6 The null hypothesis is: a)Emotional association with red is independent of gender. b)Gender is dependent upon the emotional association with red. c)The probability of selecting an emotion with red is related to gender. d)The number of children in each cell does not depend upon gender or upon emotion. e)The color red is independent of the emotion associated with it and with gender. In the paper “Color Association of Male and Female 4 th grade School Children”, children were asked to indicate what emotion they associated with the color red.

116 #7 The null hypothesis will be rejected at α =0.05 if the test statistic exceeds: a)3.84 b)5.99 c)7.81 d)9.49 e)14.07 In the paper “Color Association of Male and Female 4 th grade School Children”, children were asked to indicate what emotion they associated with the color red.

117 #8 The approximate P-value is: a)Between 0.100 and 0.900 b)Between 0.050 and 0.100 c)Between 0.025 and 0.050 d)Between 0.010 and 0.025 e)Between 0.005 and 0.010 In the paper “Color Association of Male and Female 4 th grade School Children”, children were asked to indicate what emotion they associated with the color red.


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