Nov 1 2005 ECON 240A 1 Midterm1 L Phillips Answer all five questions 1 15 points The following Box plots describe the midterm scores for Econ 240A for the past three years The total potential number of points was 75 each year Note the numbers of students taking the midterm were 45 in 2002 30 in 2003 and 35 in 2004 so note that the scale differs from box plot to box plot 2002 Smallest 34 Q1 55 Median 61 Q3 65 5 Largest 74 IQR 10 5 Outliers 34 2003 Smallest 49 Q1 59 75 Median 64 Q3 67 25 Largest 73 IQR 7 5 Outliers 2004 Smallest 18 75 Q1 34 5 Median 40 5 Q3 52 5 Largest 70 5 IQR 18 Outliers Nov 1 2005 ECON 240A 2 Midterm1 L Phillips a On average which year s class appears to do the best Explain the criterion ia that you used The class of 2003 has the highest median The question is does this mean they did the best or are differences among years obscured by the grading of different TAs etc b Which year s class was most closely bunched i e had the smallest dispersion The class of 2003 has the smallest intequartile range c Which year s class es did not have any outliers Would it have been possible given these distributions i e numbers to have an outlier at the upper end of the distribution in any of the three years The classes of 2003 and 2004 had no outliers Since Q3 1 5 IQR is the potential borderline for outliers this borderline in the various years was 2002 65 5 1 5 10 5 81 25 2003 67 25 1 5 7 5 78 5 2004 52 5 1 5 18 79 5 Every year this boundary is above the maximum score possible so no outliers at the upper end d How is an outlier calculated in 2002 Q3 1 5 IQR Q1 1 5 IQR e Do you think it would be fair to grade each year s class on an absolute scale i e based on your score as a percent of 75 points versus grading on a curve Justify your answer What can vary from year to year in addition to average student performance Comparing the median or 2004 with the other years a curve seems more appropriate See part a for a possible reason 2 15 points You conduct an experiment by throwing a fair die three times Each throw is independent a What is the probability of observing one or more sixes One or more sixes is the complement of no sixes The probability of the latter is 5 6 5 6 5 6 125 216 So the answer is 1 125 216 91 216 b What is the probability of observing exactly one six Using the binomial 3 1 2 1 6 5 6 2 75 216 c What is the probability of observing three sixes 1 6 1 6 1 6 1 216 3 15 points In last Friday s Los Angeles Times Oct 28 2005 there was a page one article on the latest poll by the Public Policy Institute of California PPIC Governor Schwarzenegger is supporting propositions 74 75 76 and 77 The fractions of 1079 likely voters in the sample that support these four propositions are 0 46 0 46 0 30 and 0 36 respectively See the attached page from the report on this poll available online a Calculate a 95 confidence interval for the population proportion of likely voters that will support proposition 74 the teacher tenure measure on election day Nov 8th if this poll is a reliable sample Using fractions round off your answer to the third decimal point Use z p hat E p hat std dev of p hat We know p hat 0 46 the expected value of p hat is p and we know std dev of p hat p hat 1 p hat 1079 1 2 0 0152 A 95 confidence interval for z is given by Nov 1 2005 ECON 240A 3 Midterm1 L Phillips the normal distribution Prob 1 96 z 1 96 0 95 Substitute in for z from above Prob 1 96 0 46 p 0 0152 1 96 Multiply the three parts of the inequality by the denominator Prob 1 96 0 0152 0 46 p 1 96 0 0152 0 95 Multiply the three parts of the inequality by minus one changing the direction of the inequality Prob 0297 p 0 46 0297 0 95 Lastly add 0 46 to all three parts of the inequality Prob 0 49 p 0 43 0 95 b This PPIC poll reports a sampling error for likely voters of plus or minus 3 Does this have any connection to your calculations in part a This is 0 0297 rounded off and converted to percent c If you were to test the null hypothesis that the population of likely voters supporting proposition is 0 5 or less against the alternative hypothesis that this population proportion is greater than 0 5 do you think you would reject the null hypothesis Explain No accept the null since the confidence interval is below 0 5 d Do you think proposition 76 the measure supporting limits on state spending is likely to pass Does our finding in class that the elasticity of state general fund expenditures with respect to personal income is significantly greater than one shed any light on these poll results for proposition 76 Won t pass People like government goods and services but are not keen on paying for them e If Governor Schwarzenegger offered you a fair bet i e even odds that proposition 74 will pass would you take the bet Yes with a 95 confidence interval below 0 5 for p this looks like a good bet 4 15 points A random sample of 100 observations was generated using the discrete Poisson probability distribution p x e x x using a procedure similar to the one in Lab Three in which we sampled from the uniform distribution The expected value of x is chosen to be three for the random number generation along with a random seed of 10 A plot of the Poisson probability distribution given a mean of three is shown in Figure 4 1 The sample observations are attached The sample mean is 3 06 The sample standard deviation is 1 99 Nov 1 2005 ECON 240A 4 Midterm1 L Phillips a Test the null hypothesis that the population mean is 3 against the alternative hypothesis that it is not equal to three Use the sample standard deviation Select a Type I error of size 5 H0 3 Ha 3 Use the t statistic t sample mean s 100 1 2 so t 3 06 3 1 99 10 0 06 0 199 0 3 a very small t statistic so accept the null b Report the numerical value of your test statistic in part a t 0 3 What is the distribution of this test statistic Student s t distribution c What is the distribution …
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