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MTH205 Introduction to Statistical Methods
Tutorial 3
Based on Chapter 3
1. State the null and alternative hypotheses to be used in testing the following claims and determine generally where the critical region is located:
(i) The mean snowfall at Lake George during the month of February is 21.8 cm.
(ii) The mean monthly household income is no more than $8000/mth.
(iii) The average rib-eye steak at the Longhorn Steak house weighs at least 340 g. .
2. In a research paper, it is claimed that mice with an average life span of 32 months will live to be about 40 months old when 40% of the calories in their diet are replaced by vitamins and protein. Is there any reason to believe that µ < 40 if 64 mice that are placed on this diet have an average life of 38 months? Assume a population standard deviation of 5.8 months. Use a p-value in your conclusion.
3. It is claimed that automobiles are driven on average no more than 20,000 km/yr. To test this claim, 100 randomly selected automobile owners are asked to keep a record of the distance they travel. Would you agree with this claim if the random sample shows an average of 23,5000 km? Assume a population standard deviation of 3900 km. Use a p-value in your conclusion.
4. A study was made to determine if the subject matter in a physics course is better understood when a lab constitutes part of the course. Students were randomly selected to participate in either a 3-semester-hour course without labs or a 4-semester-hour course with labs. In the section with labs, 11 students made an average grade of 85 with a standard deviation of 4.7, and in the section without labs, 17 students made an average grade of 79 with a standard deviation of 6.1 . It is claimed that the laboratory course increases the average grade by at least 8 points. Carry out hypothesis testing using p-value to conclude. Assume the populations to be normally distributed with equal variances.
5. A study was conducted to determine if the “strength” of a wound from surgical incision is affected by the temperature of the knife. Eight dogs were used in the experiment. “Hot” and “cold” incisions were made on the abdomen of each dog, and the strength was measured. The resulting data is given below.
Dog Knife Strength
1 Hot 5120
1 Cold 8200
2 Hot 10,000
2 Cold 8600
3 Hot 10,000
3 Cold 9200
4 Hot 10,000
4 Cold 6200
5 Hot 10,000
5 Cold 10,000
6 Hot 7900
6 Cold 5200
7 Hot 510
7 Cold 885
8 Hot 1020
8 Cold 460
(i) Write an appropriate hypothesis to determine if there is a significant di↵erence in strength between the hot and cold incisions.
(ii) Test the hypothesis using a paired t-test. Use a p-value in your conclusion.
6. Aflotoxins produced by mold on peanut crops in Virginia are to be monitored. A sample of 64 batches of peanuts reveal levels of 24.17 ppm, on average, with a variance of 4.25 ppm. Test the hypothesis that σ2 = 4.2 ppm against the alternative that σ2 = 4.2 ppm. Use a p-value in your conclusion.
7. A study is conducted to compare the lengths of time required by men and women to assemble a certain product. Past experience indicates that the distribution of times for both men and women is normal but the variance of the times for women is less than that for men. A random sample of times for 11 men and 14 women produced the following data:
Men Women
n1 = 11 n2 = 14
s1 = 6.1 s2 = 5.3
Test the hypothesis that σ1 2 = σ2 2 against the alternative that σ1 2 > σ2 2. Use 5% level of significance in your conclusion.
8. A die is tossed 180 times with the following results:
x 1 2 3 4 5 6
frequency 28 36 36 30 27 23
Is this a fair die? Use a 1% level of significance.
9. In an experiment to study the dependence of hypertension on smoking habits, the following data were taken on 180 individuals:
Non-smokers Moderate Smokers Heavy Smokers
Hypertension 21 36 30
No Hypertension 48 26 19
Test if hypertension is associated with smoking habits. Use a 5% level of significance.
10. Suppose that an allergist wishes to test the hypothesis that more than 30% of the public is allergic to some cheese products. Explain how the allergist could commit
(i) a Type I error;
(ii) a Type II error.
11. A sociologist is concerned about the e↵ectiveness of a training course designed to get more drivers to use seat belts in automobiles. What are the null and alternative hypotheses if she commits a
(i) Type I error by erroneously concluding that the training course is ine↵ective?
(ii) Type II error by erroneously concluding that the training course is e↵ective?
12. A random variable has a normal distribution with mean µ and a known variance, σ2 = 9. The null hypothesis H0 : µ = 20 is tested against the alternative hypothesis H1 : µ > 20 using a random sample of size n = 25. It is decided the null hypothesis will be rejected if the sample mean is more than 21.4 .
(i) Obtain the probability of Type I error.
(ii) Obtain the probability of Type II error and the corresponding power of this test when, in fact, µ = 21.
(iii) If we require the probability of Type I error to be 0.05 maximum, what is the new rejection rule?
(iv) What would be the probability of Type II error with the rule given in (iii) above when, in fact, µ = 21?
13. A drug for relieving nervous tension is claimed to be 60% e↵ective. However, experimental results in which the drug is administered to a random sample of 100 adults suffering from nervous tension show that only 50 received relief. Is there sufficient evidence to conclude that drug effectiveness is not 60%? Use a 5% level of significance. State the underlying assumption you use when carrying out your hypothesis test.
14. In a study to estimate the proportion of residents in a certain city and its suburbs who favor the construction of a nuclear power plant, it is found that 63 of 100 urban residents favor the construction while only 59 of 125 suburban residents are in favor. Is there a significant difference between the proportions of urban and suburban residents who favor construction of the nuclear plant? Use of a p-value to conclude.