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Chapters 23 – 25

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Percent of 15-year-olds drunk at least twice. We need a 95% confidence interval for the difference between males and females. Plan: This is matched-pairs sample, paired off by country. A histogram of the differences shows no outliers or strong skewness, so we can safely say that the distribution is plausibly normal.

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Plan: It is not clear how the individuals in the sample were selected, or if the same method of selection was used in each country, but we shall assume that the samples are at least representative of the populations they come from. In each case, we’ll assume that the samples are less than 10% of all 15-year-olds in the country.

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Mechanics and conclusion: We are 95% confident that the percentage of boys drunk at least twice is between 4.5% and 11.4% more than girls in these countries.

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The petition problem – how many valid signatures are there? a) 88.6% of the sample signatures were valid. b) In order to have the initiative certified, 82.2% of the 304,266 signatures must be valid.

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c) We start with the hypothesis that there are not enough signatures. A type I error would mean that they certify the initiative when there are not enough signatures. d) A type II error would mean that the initiative is not put on the ballot when they actually had enough signatures.

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e) Complete a significance test here: what should we use? f) To increase the power of the test, the commission could increase the sample size.

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Home burglary losses a) The margin of error is $373.50 b) They are 95% confident that the average loss in a home burglary is between $1644 and $2391. c) In repeated samples, about 95% of our confidence intervals will contain/capture/hold the true mean loss.

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Petal length of irises a) From the min, lower quartile, median, upper quartile, and max, you can make box plots… b) From the boxplots, it’s clear that versicolor has longer petals than virginica. c) The 95% C.I is (10.14, 14.46) for the difference

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Petal length of irises d) We are 95% confident that the average petal length for versicolor is between 10.14 and 14.46 mm longer than that of virginica irises. e) Since the confidence interval does not include 0, we have strong evidence of a difference in petal length.

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Speeding and racial bias a) This question calls for a one-proportion z- test, one-tailed. You should get a test- statistic of z = 4.42. There is a strong evidence that proportion of tickets given to blacks on this section of the New Jersey Turnpike is unusually high. b) Significance tests NEVER prove anything.

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Speeding and racial bias c) One statistic that would be useful is the proportion of drivers on that section of turnpike who are black.

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Frito bags a) Hypotheses: Ho: μ = 35.4Ha: μ < 35.4 b) There is one high value, but it is not technically an outlier. Otherwise, it looks roughly normal. We don’t have information on random sampling, but it’s probably a representative group. Clearly, 6 bags are less than 10% of the population.

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Frito bags c) t = 0.726 and p-value =.7497. With such a large p-value, we do NOT reject Ho. There is no evidence that the bags are underweight. d) Without the high “outlier,” we have t = -0.53 and a p-value of 0.3107, still not significant. e) The stated weight seems legitimate on these 6 bags.

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Hamsters a) We are 90% confident that the true mean litter size for hamsters is between 7.11 and 8.33 pups. b) To be more confident we need a larger interval. c) For a margin of error of 1, about 27 litters should be tested, based on t(24).

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