1. Researchers have asked several smokers how many cigarettes they had smoked in the previous day. Here are the data.

The distribution that these data are drawn from is not normal. Is there a difference between number of cigarettes smoked per day between the sexes?

U= 6, U' = 18. U_.05(2),4,6 = 22, so we can not reject H₀: Women smoke the same number of cigarettes as men.

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2. A forensic pathologist wants to know whether there is a difference between the rate of cooling of freshly killed bodies and those which were reheated, to determine whether you can detect an attempt to mislead a coroner about time of death. He tested several mice for their "cooling constant" both when the mouse was originally killed and then after the mouse was re-heated. Here's the results:

The distribution of differences is normal. Is there any difference in the cooling constants between freshly killed and reheated corpses?

Note: Because of a typo, the data as presented are not normal (the reheated value for individual 15 should read 400). The following analysis assumes normality, but uses the data above.

2-tailed, t= 1.873 We cannot reject the null hypothesis that there is a difference in the two processes' cooling coefficients.

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3. A plant ecologist wishes to test the hypothesis that the height of species X depends on the type of soil it grows in. She measures the height of 3 plants in each of 4 plots representing 4 different soil types. The results are tabulated below. (Height in centimeters.) Do the results support her hypothesis, assuming normality and equality of variances?

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4. Hanna (1953) studied hair pigment concentration in 39 pairs of monozygotic twins, Two samples were taken from each person and analyzed separately. Three readings on a spectrophotometer were taken for each sample. Here's the partial ANOVA table:

What kind of analysis has been done?

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5. Data extracted from the Canadian record book of purebred dairy cattle were used to calculate the buttermilk percentages of 5 different breeds (Ayrshire, Canadian, Guernsey, Holstein-Friesian, and Jersey) each at 2 different age classes (2-yr-old and 5-yr-old). 10 of each type of cow were used, chosen at random from the available data. If we wish to know whether the different breeds change in the same way from one age to the next, what kind of test could be done?

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6. Data were collected in Switzerland about the relationship between timing of metamorphosis and the subsequent survivorship of frogs for the first month of adult life. What test would we use to ask "Does the time of metamorphosis (expressed in days) affect the survivorship probability of adult frogs?"?

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7. Allee and Brown (1932) studied the survival time of goldfish (in minutes) when placed in colloidal silver suspensions. (Who knows why.) They used three different treatments, which differed in the concentrations of silver and other solutes. Here's a list of the survival times:

Assume that the variances of the three groups are not equal. Are the survivorship times equal in the three groups?

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8. The following temperatures were recorded in a rabbit various times after being introduced with rinderpest virus. We want to predict body temperature from knowing the time since infection of a rabbit.

a. What is the best estimate of temperature from time after injection? Calculate.

Regression. Temperature = 100 + 0.13 Time (see below)

SE_b = 0.01686

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10. Compare the assumptions of correlation with the assumptions of regression.

 Regression assumes that for all values of X that Y is normally distributed with equal variance and a random sample, but makes no assumptions about X. Correlation assumes the same things about X for all values of Y

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11. In an experiment to determine the mode of inheritance of the green mutant, 146 wild type and 30 mutant offspring were obtained when F₁ generation houseflies were crossed. test whether the data agree with the hypothesis that the ratio of wild types to mutants is 3:1.

Chi-squared = 5.94 > 3.84 so we can reject the null hypothesis that the frequency distribution is 3:1.

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12. Locality A has been sampled extensively for snakes of species S. An examination of the 167 adult males that have been collected reveals that 35 of these have pale-colored bands around their necks. From locality B, 90 miles away, we obtain a sample of 27 adult males of the same species, 6 of which show the bands. What is the chance that both samples are from the same statistical population with respect to bands?

Contingency analysis 

chi-squared = 0.022 < 3.84, so we cannot reject hypothesis of independence.

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13. In a very large Drosophila population, 30% of the flies are black and 70% are brown. Suppose that two flies are drawn at random from this population. What is the chance that these two flies are the same color? If they are the same color, what is the chance that they are both black?

 P[same color] = P[both black] + P[both brown] = 0.3² + 0.7² = 0.58

P[both black | both same color] = 0.3²/0.58 = 0.155.

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14. Consider a string of 5 random digits (that is, each digit is equally likely to be any of 0,1,2,3,…9 regardless of the other digits). What is the probability that all the digits are different?

 P[second different from first] x P[third different from first two | second different from first] x P[4th different from first three | third different from first two] x P[5th different from 1st four | 4th different from first three] =

9/10 x 8/10 x 7/10 x 6/10 = 0.3024

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15. What is the probability that if you are dealt 5 cards at random from a normal 52-card deck that you will have four-of-a kind?

Probability of one permutation which gives 4-of-a-kind = 1 x (3/51) x (2/50) x (1/49) x 1 = 4.8 x 10^-5

There are 5 permutations for each combination which gives 4-of-a-kind, so overall probability of 4-of-a-kind = 5 (4.8 x 10^-5)=2.4 x 10^-4.