This practice problem is drawn from Example 9.19 in Devore and Berk.
The fuel efficiency (mpg) of any particular new vehicle under specified driving conditions may not be identical to the EPA figure that appears on the vehicle’s sticker. Suppose that 10 different vehicles of a particular type are to be selected and driven over a certain course, after which the fuel efficiency of each one is to be determined. Let $m$ denote the true average fuel efficiency under these conditions.
Consider testing $H_{0}: \mu = 20$ versus $H_{a}: m > 20$ using the one-sample t test based on the resulting sample.
(a) Use R to produce a random sample of size 10 from a normal distribution with mean 20 and standard deviation 2. Then, conduct a t-test of this distribution against the null hypothesis that the true mean is 0. Repeat this process a large number of times (i.e. 10,000), saving the resulting p-value each time. What proportion of these simulations would reject the null hypothesis at an $\alpha = 0.05$ critical value? If you produce a plot of this distribution of these p-values, what do you observe? (b) Use R to produce a random sample of size 10 from a normal distribution with mean 21 and standard deviation 2. Then, conduct a t-test of this distribution against the null hypothesis that the true mean is 0. Repeat this process a large number of times (i.e. 10,000), saving the resulting p-value each time. What proportion of these simulations would reject the null hypothesis at an $\alpha = 0.05$ critical value? If you produce a plot of this distribution of these p-values, what do you observe? (c) Use R to produce a random sample of size 10 from a normal distribution with mean 22 and standard deviation 2. Then, conduct a t-test of this distribution against the null hypothesis that the true mean is 0. Repeat this process a large number of times (i.e. 10,000), saving the resulting p-value each time. What proportion of these simulations would reject the null hypothesis at an $\alpha = 0.05$ critical value? If you produce a plot of this distribution of these p-values, what do you observe?
Scientists have recently become concerned about the safety of Teflon cookware and various food containers because perfluorooctanoic acid (PFOA) is used in the manufacturing process. An article in the July 27, 2005, New York Times reported that of 600 children tested, 96% had PFOA in their blood. According to the FDA, 90% of all Americans have PFOA in their blood. (a) Does the data on PFOA incidence among chil- dren suggest that the percentage of all children who have PFOA in their blood exceeds the FDA percentage for all Americans? Carry out an appropriate test of hypotheses. (b) If 95% of all children have PFOA in their blood, how likely is it that the null hypothesis tested in (a) will be rejected when a signifi- cance level of .01 is employed? (c) Referring back to (b),what sample size would be necessary for the relevant probability to be .10?
A random sample of soil specimens was obtained, and the amount of organic matter (%) in the soil was determined for each specimen, resulting in the accompanying data (from “Engineering Properties of Soil,” Soil Sci., 1998: 93–102).
Notice that these values are provided in percentage terms (i.e. what is noted as 1.10 is 1.1%)
dirt <- c(1.10, 5.09, 0.97, 1.59, 4.60,
0.32, 0.55, 1.45, 0.14, 4.47,
1.20, 3.50, 5.02, 4.67, 5.22,
2.69, 3.98, 3.17, 3.03, 2.21,
0.69, 4.47, 3.31, 1.17, 0.76,
1.17, 1.57, 2.62, 1.66, 2.05 )
(a) Create a plot of this dirt data that examines the distribution of this sample. Does a t-test seem appropriate in this case? Why or why not?
(b) Compute the values for the sample average, sample standard deviation, and standard error of the mean.
(c) Either by hand (eww...dirty), or using the built-in function t.test conduct a test for whether this sample is drawn from a population with a percentage of organic matter equal to 3%. Because we have only a few samples, suppose that you would be willing to reject the null hypothesis if a p-value is smaller than or equal to 0.10.
Does anyone else share this odd facination: One of life's small joys for Alex is keeping a pen for long enough that you write all of the ink out of it.
A pen has been designed so that true average writing lifetime under controlled conditions (involving the use of a writing machine) is at least 10 h. A random sample of 18 pens is selected, the writing lifetime of each is determined, and a normal probability plot of the resulting data sup- ports the use of a one-sample t test. (a) What hypotheses should be tested if the investigators believe a priori (aka ahead of time) that the design specification has been satisfied? (b) What conclusion is appropriate if the hypotheses of part (a) are tested, $t = -2.3$, and $\alpha = 0.05$? (c) What conclusion is appropriate if the hypotheses of part (a) are tested, $t = -1.8$, and $\alpha=.01$? (d) What should be concluded if the hypotheses of part (a) are tested and $t = -3.6$?