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In each class, the number (#) of data values expected to occur, according to the fitted distribution, is simply the probability of occurrence in that class multiplied by the sample size, n. (5.14) χ 2 = ∑ classes # Observed − # Expected 2 # Expected = ∑ classes # Observed − n Pr data in class 2 n Pr data in class. It should be noted that the test is two-tailed (in contrast to the majority of cases where χ 2-tests are used) since values significantly smaller or larger than n−1 can lead to rejection of the null hypothesis. A χ 2-test can be used for testing whether s 2/ x̄ deviates significantly from 1 since χ 2= ( n−1) s 2/ x̄ with n−1 degrees of freedom. Especially if the sample size n is small, s 2/ x̄ will exhibit large variation due to sampling noise. However, since data originate from sampling, they will always be associated with some variation, so it is likely that some deviation in s 2/ x̄ from unity will occur even if the underlying distribution is random. If the ratio exceeds 1, it indicates that the population has a patchy (or clumped) distribution whereas a value less than unity indicates an even (or regular) distribution. If this is the case, it is expected that the variance should equal the average so that the ‘index of dispersion’, s 2/ x̄, is approximately equal to 1. Nachman, in Encyclopedia of Ecology (Second Edition), 2008Īs the null hypothesis, it is assumed that individuals in a population are randomly distributed among the n sampling units of a sample. Do you have any concerns about the experimental design or the analysis?ĭ) (Optional) Perform a multiple comparisons test to determine which drugs differ in terms of mean pain level reported.M.K. What can you conclude?Ĭ) Check the assumptions and conditions for an ANOVA.
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\hline \text & 5Ī) State the null and alternative hypotheses, being careful to talk about Drug and Pain levels as well as parameters.ī) Perform an ANOVA on these data.
#STATS MODELING THE WORLD CHAPTER 26 ANSWERS DOWNLOAD#
To talk about download Time and Time of Day as well He downloaded the file 48 times in all, 16 times at each Time of $D a y,$ and recorded the Time in seconds that the download took.Ī) State the null and alternative hypotheses, being careful He placed a file on a remote server and then proceeded to download it at three different time periods of the day. To see how much of a difference time of day made on the speed at which he could download files, a college sophomore performed an experiment. What does it say about the questions in part c? For each pair of shelves, the difference is shown along with its standard error and significance level. Does sugar content vary by shelf? At the top of the next column is a boxplot and an ANOVA table for the 77 cereals.Ī) What are the null and alternative hypotheses?ī) What does the ANOVA table say about the null hypothesis? (Be sure to report this in terms of Sugars and Shelves.)Ĭ) Can we conclude that cereals on shelf 2 have a higher mean sugar content than cereals on shelf $3 ?$ Can we conclude that cereals on shelf 2 have a higher mean sugar content than cereals on shelf $1 ?$ What can we conclude?ĭ) To check for significant differences between the shelf means, we can use a Bonferroni test, whose results are shown below. We have data on the shelf as well as the sugar, sodium, and calorie content of 77 cereals. Supermarkets often place similar types of cereal on the same supermarket shelf. $ What does this say about the null hypothesis of equal means?ĭ) What assumptions have you made in order to answer part c?Į) What would you like to see in order to justify the conclusions of the $F$ -test?į) What is the average size of the error standard deviation in particulate emissions? A partially completed Analysis of Variance table of the data follows on the next page.Ī) Calculate the mean square of the treatments and the mean square of the error.ī) Form the $F$ -statistic by dividing the two mean squares.Ĭ) The P-value of this $F$ -statistic turns out to be $0.0000949. For each run, the same material was produced, and the particulate emissions coming out of the scrubber were measured (in parts per billion). An experiment to determine which smokestack scrubber design is best was run by placing four scrubbers of different designs on an industrial stack in random order. One way to reduce the pollution is to put a filter, or scrubber, at the end of the smokestack to trap the particulates. Particulate matter is a serious form of air pollution often arising from industrial production.