HOMEWORK - T-TEST & HYPOTHESIS TESTING PROBLEMS
Problem 1. Six healthy three-year-old female Suffolk sheep were injected with the antibiotic Gentamicin, at a dosage of 10 mg/kg body weight. Their blood serum concentration (µg/ml) of Gentamicin 1.5 hours after the injection were as follows:
33, 26, 34, 21, 23, 25
(a) Construct a 90% confidence interval for the population mean. Be sure to interpret this confidence interval in the context of this setting.
(b) Construct a 99% confidence interval for the population mean. Be sure to interpret this confidence interval in the context of this setting.
Problem 2. Modern cars have measurements of the miles per gallon (mpg) available to drivers in a dash board display. However, some people still prefer to hand compute the mpg of their own cars by calculating the miles driven between gas fill-ups along with the amount of gas purchased at each gas station during the fill up. The following data are the differences between the drivers own calculations and the car computer's for a random sample of 20 records.
(a) Load the following 20 data points into R and calculate the sample mean x¯ and the standard deviation S using the following R commands.
x = c(5.0, 6.5, -0.6, 1.7, 3.7, 4.5, 8.0, 2.2, 4.9, 3.0, 4.4, 0.1, 3.0, 1.1, 1.1, 5.0, 2.1, 3.7, -0.6, -4.2)
xbar=mean(x)
s=sd(x)
What is the sample mean ¯x and the standard deviation S?
(b) You want to test the hypothesis that the population mean difference between the computer and hand calculations is equal to zero. State what the null and alternate hypotheses using µ as the true mean difference. (Keep in mind that this is a two-sided hypothesis test).
(c) Calculate what the test statistic T∗ = √n(x¯-µ0)/S is for this data set.
(d) Draw a bell curve centered at 0 and label this curve the null distribution. Annotate on your drawing where the test statistic T∗ falls using a line. What is the null distribution?
(e) Shade in the area under the bell curve which corresponds to the p-value. Keep in mind that this is a two-sided hypothesis test.
(f) Calculate the p-value by calculating the area under the curve that you shaded in your diagram.
Problem 3. Some soap manufacturers sell special "antibacterial" soaps. However, one might expect ordinary soap to also kill bacteria. To investigate this, a researcher prepared a solution from ordinary, non-antibiotic soap and a control solution of sterile water. The two solutions were placed onto petridishes and E. coli bacteria were added. The dishes were incubated for 24 hours and the number of bacteria colonies on each dish were counted. The data are given in the following table.
|
|
Control Group X
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Soap Group Y
|
|
|
30
|
76
|
|
|
36
|
27
|
|
|
66
|
16
|
|
|
21
|
30
|
|
|
63
|
26
|
|
|
38
|
46
|
|
|
35
|
6
|
|
|
45
|
|
|
n
|
8
|
7
|
|
sample mean
|
41.8
|
32.4
|
|
sample sd
|
15.6
|
22.8
|
(a) Construct a 90% confidence interval for the above data. Be sure to interpret this confidence interval in the context of this setting.
A test should be performed to determine whether soap more effective than the control. Conduct the hypothesis test at the 5% level.
(b) State the null and alternative hypotheses in words and symbols.
(c) Compute the test statistic
(d) Compute the P-value
(e) State the conclusion of the test in the context of this setting.
Problem 4. A certain manufactured product is supposed to contain at least 23% potassium by weight. A sample of 10 specimens of this product had an average percentage of 23.2 with a standard deviation of 0.2. If the mean percentage is found to be less than 23, the manufacturing process will be recalibrated.
(a) State the appropriate null and alternative hypotheses.
(b) Compute the test statistic.
(c) Compute the P-value
(d) State the conclusion of the test in the context of this setting.
Problem 5. Surfactants are chemical agents, such as detergents, that lower the surface tension of a liquid. Surfactants play an important role in the cleaning of contaminated soils. In an experiment to determine the effectiveness of a certain method for removing toluene from sand, the sand was washed with a surfactant, and then rinsed with de-ionized water. Of interest was the amount of toluene that came out in the rinse.
In five such experiments, the amounts of toluene removed in the rinse cycle, expressed as a percentage of the total amount originally present, were 5.0, 4.8, 9.0, 10.0, and 7.3.
(a) Find a 95% confidence interval for the percentage of toluene removed in the rinse.
(b) Conduct a hypothesis at the α = 5% level. Can you conclude that the mean amount of toluene removed in the rinse is less than 8%?
Problem 6. In a study of the relationship of the shape of a tablet to its dissolution time, 6 disk-shaped ibuprofen tablets and 8 oval-shaped ibuprofen tablets were dissolved in water. The dissolve times, in seconds, were as follows:
Disk: 269.0, 249.3, 255.2, 252.7, 247.0, 261.6
Oval: 268.8, 260.0, 273.5, 253.9, 278.5, 289.4, 261.6, 280.2
Can you conclude that the mean dissolve times differ between the two shapes?
Conduct a hypothesis test at the α = 5% level.
(a) State the appropriate null and alternative hypotheses.
(b) Compute the test statistic.
(c) Compute the P-value
(d) State the conclusion of the test in the context of this setting.
Problem 7. Two formulations of a certain coating, designed to inhibit corrosion, are being tested. For each of eight pipes, half the pipe is coated with formulation A and the other half is coated with formulation B. Each pipe is exposed to a salt environment for 500 hours. Afterward, the corrosion loss (in µm) is measured for each formulation on each pipe.
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Pipe
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A
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B
|
|
1
|
197
|
204
|
|
2
|
161
|
182
|
|
3
|
144
|
140
|
|
4
|
162
|
178
|
|
5
|
185
|
183
|
|
6
|
154
|
163
|
|
7
|
136
|
156
|
|
8
|
130
|
143
|
Can you conclude that the mean amount of corrosion differs between the two formulations?
Conduct a hypothesis test at the α = 10% significance level.
(a) State the appropriate null and alternative hypotheses.
(b) Compute the test statistic.
(c) Compute the P-value
(d) State the conclusion of the test in the context of this setting.
Problem 8. A 95% confidence interval for µX -µY is (-0.3, 0.15). Based upon the data from which the confidence interval was constructed, someone wants to test H0: µX = µY versus HA : µX ≠ µY. at the α = 5% significance level.
(a) Based upon the confidence interval, what is your conclusion of the hypothesis test? (Explain)
(b) Can we use the above confidence interval to conduct the hypothesis test at the α = 10% level? Why or why not?
Problem 9. Suppose we have conducted a t-test for the differences of mean µX -µY, with α = 0.10, and the P-value is 0.07. For each the following statements, say where the statement is true or false and explain why.
(a) We reject H0 with α = 0.10.
(b) We have significant evidence for HA with α = 0.10.
(c) We would reject H0 if α were 0.05 instead.
(d) The probability that X- is greater than Y- is 0.07.
(e) There is a 7% probability that H0 is true.
(f) If H0 is true, the probability of getting a test statistic at least as extreme as the value of ts that was actually obtained is 7%.
Problem 10. Myocardial blood flow (MBF) was measured for two groups of subjects after five minutes of bicycle exercise. The normoxia ("normal oxygen") group was provided normal air to breathe whearas the hypoxia group was provided with a gax mixture with reduced oxygen to simulate high altitude. The results (ml/min/g) are shown in the table below.
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NORMOXIA X
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HYPOXIA Y
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|
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3.45
|
6.37
|
|
|
3.09
|
5.69
|
|
|
3.09
|
5.58
|
|
|
2.65
|
5.27
|
|
|
2.49
|
5.11
|
|
|
2.33
|
4.88
|
|
|
2.28
|
4.68
|
|
|
2.24
|
3.50
|
|
|
2.17
|
|
|
|
1.34
|
|
|
n
|
10
|
8
|
|
sample mean
|
2.51
|
5.14
|
|
sample sd
|
0.60
|
0.84
|
We wish to investigate the effect of hypoxia on MBF.
(a) Construct a 90% confidence interval for the differences of the mean MBF for the two groups, i.e. µX -µY.
(b) Use the above confidence interval to conduct a hypothesis test with α = 0.10.