STT 231 – 001
PRACTICE EXERCISES ON STATISTICAL INFERENCES
· SAMPLE DISTRIBUTION FOR ONE SAMPLE PROPORTION
QUESTIONS 1 – 2
Information on a packet of seeds claims that the germination rate is 84%. The packet contains 150 seeds. Let represent the proportion of seeds in the packet that will germinate.
1. The sampling distribution model for is
A. N(0.84, 0.0299) B. N(0.16, 0.0299) C. N(0.84, 0.0001) D. N(0.0215, 0.92)
2. What’s the approximate probability that more than 90% of the 150 seeds in the packet will germinate?
A. 0.9778 B. 0.0224 C. 0.0299 D. 0.90 E. 0.16
3. When a truckload of apples arrives at a packing plant, a random sample of 180 is selected and examined for bruises, discoloration, and other defects. The whole truckload will be rejected if more than 4% of the sample is unsatisfactory. Suppose that in fact 7% of the apples on the truck do not meet the desired standard. What’s the probability that the shipment will be accepted anyway?
A. 0.0571 B. 0.981 C. – 1.578 D. 0.943 E. 0.0409
QUESTIONS 4 – 5
It’s believed that 5% of children have a gene that may be linked to juvenile diabetes. Researchers hoping to track 30 of these children for several years test 742 newborns for the presence of this gene.
4. The sampling distribution of the proportion of children with gene linked to juvenile diabetes is best described as
A. N(0.04, 0.0072) B. N(0.0072, 0.04) C. N(0.008, 0.05) D. N(0.05, 0.008)
5. What is the probability that they find enough subjects for their study?
A. 0.0404 B. 0.0072 C. 0.960 D. 0.1170 E. 0.884
6. Based on past experiences, a bank believes that 8% of the people who receive loans will not make payments on time. The bank has recently approved 200 loans. What’s the probability that over 13% of these clients will not make timely payments?
A. 0.9953 B. 0.0192 C. 0.0046 D. 0.92 E. 0.261
QUESTIONS 7 – 9
Suppose that 65% of the adult residents in North Dakota favor the death penalty. In a simple random sample of 100 North Dakota adult residents, let denote the proportion that favor the death penalty.
7. What is the mean (expected) value of ?
(a) 50 (b) 60 (c) 65 (d) 0.50 (e) none of these
8. What is the standard deviation of ?
(a) 0.15 (b) 0.048 (c) 5 (d) 0.09 (e) none of these
9. What is the approximate probability that 57% or less of the sample residents favor death penalty, that is, what is P(£0.57)
(a) 0.50 (b) 0.40 (c) 0.28 (d) 0.16 (e) 0.05
QUESTIONS 10 – 11
Information on a packet of seeds claims that the germination rate is 92%. The packet contains 160 seeds. Let represent the proportion of seeds in the packet that will germinate.
10. The sampling distribution model for is
A. N(0.92, 0.0215) B. N(0.08, 0.0215) C. N(147.2, 12.8) D. N(0.0215, 0.92)
11. What’s the approximate probability that more than 95% of the 160 seeds in the packet will germinate?
A. 0.92 B. 0.081 C. 0.0215 D. 0.95
12. When a truckload of apples arrives at a packing plant, a random sample of 150 is selected and examined for bruises, discoloration, and other defects. The whole truckload will be rejected if more than 5% of the sample is unsatisfactory. Suppose that in fact 8% of the apples on the truck do not meet the desired standard. What’s the probability that the shipment will be accepted anyway?
A. 0.0222 B. 0.9778 C. 0.088 D. 0.912
QUESTIONS 13 – 14
It’s believed that 4% of children have a gene that may be linked to juvenile diabetes. Researchers hoping to track 20 of these children for several years test 732 newborns for the presence of this gene.
13. The sampling distribution of the proportion of children with gene linked to juvenile diabetes is best described as
A. N(0.04, 0.0072) B. N(0.0072, 0.04) C. N(0.027, 0.04) D. N(0.027, 0.0072)
14. What is the probability that they find enough subjects for their study?
A. 0.0273 B. 0.0072 C. 0.960 D. 0.04
· SAMPLING DISTRIBUTION FOR TWO SAMPLE PROPORTIONS
1. There has been debate among doctors over whether surgery can prolong life among men suffering from prostrate cancer, a type of cancer that typically develops and spreads very slowly. In the summer of 2003, The New England Journal of Medicine published results of some Scandinavian research. Men diagnosed with prostrate cancer were randomly assigned to either undergo surgery or not. Among the 347 men who had surgery, 16 eventually died of prostrate cancer, compared with 31 of the 348 men who did not have surgery. What is the standard error of the difference in the two proportions?
A. 0.0189 B. 0.00036 C. 0.0113 D. 0.037 E. 0.08908
2. Researchers at the National Cancer Institute released the results of a study that investigated the effect of weed – killing herbicides on house pets. They examined 827 dogs from homes where an herbicide was used on a regular basis, diagnosing malignant lymphoma in 473 of them. Of the 130 dogs from homes where no herbicides were used, only 19 were found to have lymphoma. What’s the standard error of the difference in the two proportions?
A. 0.0354 B. 0.0013 C. 0.0172 D. 0.0309 E. 0.4258
3 - 4. A veterinarian wants to compare the rate of hip dysplasia in Boxers (breed of dog) and the rate hip dysplasia in American Bulldogs (breed of dog). A random sample of 80 Boxers has 8 dogs with hip dysplasia. A random sample of 140 American Bulldogs has 21 dogs with hip dysplasia.
3. From these two samples, the difference between 2 sample proportions, is
(a) 0.1 (b) 0.15 (c) -0.05 (d) 0.8 (e) -0.13
4. The standard error of the sampling distribution of is
(a) .085 (b) .071 (c) .045 (d) 0.037 (e) .101
5 - 7: At a small college somewhere on the East coast, 20% of the girls and 30% of the boys smoke at least once a week. Independent random samples of 75 girls and 100 boys are to be selected and the proportions of smokers in the samples are to be calculated.
5. What is the mean of the sampling distribution of the difference in the sample proportion of girl smokers and the sample proportion of boy smokers?
KEY: –0.10
6. What is the standard deviation of the sampling distribution of the difference in the sample proportion of girl smokers and the sample proportion of boy smokers?
KEY: 0.0651
7. What is probability that the sample proportion of girl smokers is greater than the sample proportion of boy smokers?
KEY: 0.0623
8 - 9: In the Youth Risk Behavior Survey (a study of public high school students), a random sample showed that 45 of 675 girls and 103 of 621 boys had been in a physical fight on school property one or more times during the past 12 months.
8. What is the difference in sample proportions of students who had been in a fight (boys – girls)?
KEY: 0.0992.
9. What is the standard error of the difference in sample proportions?
KEY: 0.0177
10 - 12: Suppose that 60% of all teenagers — both boys and girls — are classified as having good grades. Independent random samples of 537 boys and 689 girls who go to school in the Washington, DC, area are to be selected and surveyed and the proportions of teenagers with good grades are to be calculated.
10. What is the mean of the sampling distribution of the difference between the two sample proportions
(boys – girls)?
KEY: 0
11. What is the standard deviation of the sampling distribution of the difference between the two sample proportions (boys – girls)?
KEY: 0.0282
12. What is the probability that the difference between the two sample proportions (boys – girls) is greater than 5 percentage points (0.05)?
KEY: 0.0381
13 - 15: Suppose that 80% of all English majors and 85% of all engineering majors at a Minnesota college wear winter boots when there is snow on the ground. Two independent random samples of 40 English majors and 60 engineering majors are to be selected during a day with snow on the ground and the proportions of students with winter boots are to be calculated.
13. What is the mean of the sampling distribution of the difference between the two sample proportions
(English majors – engineering majors)?
KEY: −0.05
14. What is the standard deviation of the sampling distribution of the difference between the two sample proportions (English majors – engineering majors)?
KEY: 0.0783
15. What is the probability that more English majors than engineering majors in the sample wear winter boots?
KEY: 0.2616
16 - 17: A survey at a large public university reveals that many students hold a (part-time) job while going to college. Out of 127 female students surveyed, 95 have a job and out of 143 male students, 97 have a job.
16. What is the estimate for the difference between the proportions of female and male students who have a job while going to college?
KEY: 0.0697
17. What is the standard error of the estimate?
KEY: 0.0549
18. Suppose that the mean of the sampling distribution for the difference in two sample means is 0. This tells us that
A. the two sample means are both 0.
B. the two sample means are equal to each other.
C. the two population means are both 0.
D. the two population means are equal to each other.
KEY: D
· SAMPLING DISTRIBUTION FOR ONE SAMPLE MEAN
· SAMPLING DISTRIBUTION FOR TWO SAMPLE MEANS
QUESTIONS 1 – 4
Statistics from Cornell’s Northeast Regional Climate Center indicate that Ithaca, NY, gets an average of 35.4 inches of rain each year, with a standard deviation of 4.2 inches.
1. During what percentage of years does Ithaca get more than 30 inches of rain?
A. approximately 10.03% B. approximately 10.95%
C. approximately 90.07% D. approximately 5.82
E. approximately 94.18%
2. Less than how much rain falls in the driest 25% of all years?
A. 13.7 inches B. 10.03 inches C. 10.4 inches D. 32.6 inches.
A Cornell University student is in Ithaca for 4 years. Let represent the mean amount of rain for those 4 years.
3. The sampling distribution model of this mean, , is best described as
A. N(35.4, 4.2) B. N(4.2, 35.4) C. N(35.4, 2.1) D. N(2.1, 35.4)
4. What’s the probability that those 4 years average less than 30 inches of rain?
A. 0.995 B. 0 C. 0.11 D. 0.005
QUESTIONS 5 – 7
The weight of potato chips in a medium-size bag is stated to be 10 ounces. The amount that the packaging machine puts in these bags is believed to have a Normal model with mean 10.2 ounces and standard deviation 0.12 ounces.
5. What fraction of all bags sold are underweight?
A. 0.9804 B. 0.0478 C. 0.863 D. 0.2
6. Some of the chips are sold in “bargain packs” of 3 bags. What’s the probability that none of the 3 is underweight?
A. 0.863 B. 0.1434 C. 0.000109 D. 0.9522 E. None of these
7. What’s the probability that the mean weight of the 3 bags is below the stated amount?
A. 0.069 B. 0.0478 C. 0.9522 D. 0.0019
8. The average composite ACT score for Ohio students who took the test in 2003 was 21.4. Assume that the standard deviation is 1.05. In a random sample of 25 students who took the exam in 2003, what is the probability that the average composite ACT score is 22 or more? (HINT: Make sure to identify the sampling distribution you use and check all necessary conditions before you proceed to solve the problem.)
A. 0.21 B. 2.86 C. 0.0021 D. 0.9979 E. 0.79
QUESTIONS 9 – 12
Statistics from Cornell’s Northeast Regional Climate Center indicate that Ithaca, NY, gets an average of 35.4 inches of rain each year, with a standard deviation of 4.2 inches.
9. During what percentage of years does Ithaca get more than 40 inches of rain?
A. approximately 13.7% B. approximately 10.95% C. approximately 31.2%
10. Less than how much rain falls in the driest 20% of all years?
A. 13.7 inches B. 10.95 inches C. 39.6 inches D. 31.9 inches.
A Cornell University student is in Ithaca for 4 years. Let y represent the mean amount of rain for those 4 years.
11. The sampling distribution model of this mean, y is best described as
A. N(35.4, 4.2) B. N(4.2, 35.4) C. N(35.4, 2.1) D. N(2.1, 35.4)
12. What’s the probability that those 4 years average less than 30 inches of rain?
A. 0.995 B. 0 C. 0.11 D. 0.005
13. Assume that the duration of human pregnancies can be described by a Normal model with mean 266 days and standard deviation 16 days.
(a) What percentage of pregnancies should last between 270 and 280 days?
(b) At least how many days should the longest 25% of all pregnancies last? 276.8 days
(c) Suppose a certain obstetrician is currently providing prenatal care to 60 pregnant women. Let y – bar represent the mean length of their pregnancies. According to the Central Limit Theorem, what’s the distribution of this sample mean, y – bar? Specify the model, mean, and standard deviation. Mean = 266 days; Standard deviation = 2.07 days
(d) What’s the probability that the mean duration of these patient’s pregnancies will be less than 260 days? 0.002
14. Carbon monoxide (CO) emissions for a certain kind of car vary with mean 2.9 g/mi and standard deviation 0.4 g/mi. A company has 80 of these cars in its fleet. Let y – bar represent the mean CO level for the company’s fleet.
(a) What’s the approximate model for the distribution of y – bar? Explain. N(2.9, 0.045)
(b) Estimate the probability that y – bar is between 3.0 and 3.1 g/mi.
(c) There is only a 5% chance that the fleet’s mean CO level is greater than what value? 2.97 g/mi
15. A fourth grade class of 28 students is given a standardized math test. The mean score of the 12 boys is 25 with a standard deviation of 3. The mean score of the 16 girls is 24 with a standard deviation of 4. What is the standard error for the sampling distribution of ?