Problems: Confidence Intervals Part
2
11. Suppose we are testing a make of car to determine its gas mileage. We know that miles per gallon is normally distributed with a standard deviation of one mile per gallon. (Never mind how we know this--we just do.)
- a) Taking a random sample of 16 cars, we find the sample mean is 24 miles per gallon. Find the interval in which we are 95% certain that the true mean lies.
- b) Suppose we do not know the standard deviation but estimate it from the sample, finding that the standard deviation of the 16 cars is one mile per gallon. Find the interval in which we are 95% certain the true mean lies.
12. Suppose the weights of oranges from the trees in a grove of orange trees are normally distributed.
- a) If a random sample of sixteen is taken and the sample mean is 3.2 ounces with a sample standard deviation of .8 ounces, what is the interval within which we are 90% confident that the true mean lies?
- b) Suppose that instead of using the t-tables as you should have in the previous question, you used the normal tables. What answer would you get? Why are these answers different?
13. Suppose an investigator wants to know the average age of automobiles in Indiana.
- a) Would the distribution of car ages be normal, uniform, binomial, or some other distribution? Is it a discrete or continuous distribution?
- b) Suppose that the investigator takes a random sample of 100 cars. What distribution will the sample mean have? Or is the sample mean a fixed number? Explain.
- c) What is the expected value of the sample mean? Why is this important?
- d) Suppose the investigator finds a sample mean of 4.4 years and a sample standard deviation of 2.1 years. (With a sample this large the sample standard deviation is a fairly reliable estimate of the population standard deviation and may be used in its place.) What is a 95% confidence interval for the true average age of Indiana cars? (Data are all hypothetical.)
14. A random sample of 64 bricks weighs 50.28 ounces with a standard deviation of .8 ounces.
- a) What is the 95% confidence interval for the true average weight of the bricks produced by this process?
- b) What does this interval mean?
15. An agronomist is interested in the yield of a newly developed tomato variety. A sample of 25 plants gives an average yield of 34 pound during a 120-day growing season, with a sample standard deviation of three pound. What is the 95% confidence interval for the true average yield of this variety?
16. A politician facing an upcoming election wants to know what proportion of a population favors him over his opponent. He hires a pollster to give him a result that is accurate within .02 (or 2%) with 95% confidence. Assuming that this is a close race, how large a sample must the pollster take? (Hint: in finding the standard error of the proportion, use p=.5)
17. Suppose a lumber company is interested in purchasing 10,000 acres of forest. If the trees have an average diameter of 2.5 feet and a standard deviation of .5 feet,
- a) what percentage of frees will have a diameter between two and three feet if tree diameters are normally distributed?
- b) what percentage will have a diameter between three and four feet?
- c) If two trees are picked at random, what is the probability that they will both have diameters greater than three feet?
- d) Suppose this company does not know the average diameter of the trees, but does know that the standard deviation is one foot. They intend to estimate the true mean with a sample mean. How large a random sample do they need if they want to be 95% confident that they will estimate the true mean within three inches?
- e) How large a random sample do they need if they want to be 95% confident that they will estimate the true mean within one inch?
- f) The company is also interested in how many trees are in the forest. To estimate this they select five one-acre plots at random and count the trees on each plot. Why is it important that the plots be selected randomly?
- g) Suppose they get this result:
- Plot A: 40 trees
- Plot B: 46 trees
- Plot C: 36 trees
- Plot D: 39 trees
- Plot E: 39 trees
- What is the arithmetic mean of this sample?
- h) What is the median?
- i) What is the variance?
- j) What is the standard deviation?
- k) Suppose from past experience the company is certain that the standard deviation of trees per acre in a forest is four trees. How confident are they that there are at least 36 trees per acre in this forest based on the above sample?
18. The average SAT verbal score of a random sample of 25 sophomores from a large college was 394 with a standard deviation of 75. What is:
- a) a 90% confidence interval for the entire population of college sophomores at this college?
- b) a 95% confidence interval for the entire population of college sophomores at this college?
- c) Approximately how much confidence would we have in the interval 394 ± 20?
- d) How big would our sample size have to be to have a 95% confidence interval that is ± 20?
19. The following IQ scores have been obtained from a random sample of persons from a large population: {70, 110, 110, 110, 110, 80, 120, 110, 90}. Construct a 95-percent confidence interval estimate of the population mean IQ.
20. Suppose that the average weight of widgets is 130 grams. Taking repeated samples of 25 observations, we find that 95% of the time the sample mean will fall in the interval 120 to 140.
- a) Approximately what is the standard deviation of the original population?
- b) How large would our sample size have to be if we wanted 95% of our observations to fall in the interval 125 to 135?
21. A random sample of 400 fifth grades finds that they eat an average of 1850 calories per day. The standard deviation of the sample is 250 calories. What is a 95% confidence interval for the average amount of calories that all fifth graders eat?
22. A sample of 16 light bulbs is tested to see how bright they burn. The average of the sample is 2000 lumens with a sample standard deviation of 48 lumens.
- a) What is the 99% confidence interval for the true mean?
- b) What is the 90% confidence interval for the true mean?
23. A sample of observations has been take and using SPSS, the following results are obtained.
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One |
141 |
510.5332 |
252.5299 |
21.3511 |
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Two |
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One |
23.911 |
140 |
.000 |
510.5332 |
468.3210 |
552.7454 |
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Two |
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- a) Compute a 90% confidence interval for the first variable.
- b) Compute a 99% confidence interval for the second variable.
- c) Suppose we wanted to test the hypothesis that the true mean of the first variable was 550. Could we reject this if we set alpha = .05? Compute the appropriate t-value and explain what it tells you. (This part of the question requires knowledge of hypothesis testing, the topic of the next section.)
24. A test of 25 cigarette lighters show that they average 2947 lights with a standard deviation of 250 lights.
- a) Using the t-tables, construct a 95% confidence interval for the population mean.
- b) Using the normal tables, construct a 95% confidence interval for the population mean. How do you explain the difference in your result from part a?
- c) Using the Normal table, construct a 98% confidence interval for the population mean.
- d) Approximately how much confidence would you have in the interval 2947 ± 50?
- e) How large must the sample be if want a precision of ± 50 with 95% confidence?
26. A random sample of 100 ears of corn from a field shows that they weigh an average of .82 pounds with a sample standard deviation of .12 pounds.
- a) What is a 94 confidence interval for the true mean weight?
- b) What does this answer mean?
- c) Suppose the person who took the sample and computed the results did not know much about statistics and he forgot to include the sample mean. Instead he only reported that the median of the sample was .81. If the original distribution of the weight of corn is symmetric, the population mean and the population median are the same, and the sample median can be used as an estimate of the population mean. However, as we take sample after sample, the median of the samples will bounce around more than the mean--the sample median will have a distribution that has a bigger standard deviation than the distribution of the sample mean. Some brilliant mathematician showed that the distribution of the sample median will have a standard deviation that is larger than the standard error of the mean by a factor of the square root of one half of pi. What would a 94% confidence interval of the true mean be if the sample median was used as the estimator of the population mean? (Note: this question is highlighting that one of the nice properties of the mean compared to the median is that it is more efficient--that is, it bounces around less from sample to sample than the sample median does.)
27. Congressman Friendly mails a questionnaire to his constituents to learn their views on an energy issue. Can he draw valid conclusions about how all of his constituents view this issue from the results of his survey? Explain. If he cannot draw valid conclusions, why would he do the survey?
28. A random sample of 1600 people is taken to obtain their view on the economy. 67% say they expect things to get worse. What is a 95% confidence interval for the entire population?
29. A garden seed distributor wants to test the germination of a shipment of potato seeds. (Excuse the attempted humor.) Previous experience has shown that he can expect between 75% and 98% of them to germinate. How large a sample must he take if he wants to estimate the percentage of this shipment that will germinate within ± .05 with 05% confidence?
30. A garden seed distributor wants to test the germination rate of a shipment of onion seed. He has no idea of what the rate may be. How large a sample must he take if he wants to estimate the percentage of this shipment that will germinate within ± .05 with 95% confidence?
31. Suppose that a retail store is interested in knowing the average age of their inventory. They take a random sample of 400 items and find that the average age is 98 days with a standard deviation of 38 days.
- a) What is a 90% confidence interval for the population mean in this case?
- b) Suppose that they found that the sample median was 96 days. Find a 90% confidence interval for the population mean using the sample median as a point estimate instead of the sample mean. (For this we need to assume that the original population was distributed as a normal distribution, in which case the mean and median are identical. However, the standard error of the sample median will be larger than the standard error of the standard error of the sample mean by a factor of 1.2533, which is the square root of one-half of pi.)
- c) If two estimators are unbiased (they tend not to overpredict or underpredict), the one whose sampling distribution has the smaller variance is more efficient. The sample median is an unbiased estimator of the sample mean if the original population is distributed normally, but as mentioned in part b, it has a larger standard deviation. What must its distribution look like compared to the distribution of the sample mean? (Assume that the curve below shows the distribution of the sample mean form sample of sample. draw on an appropriate curve showing what the distribution of the sample median would look like.)
- d) Both the sample mean and the sample median are consistent estimators of the sample mean if the original distribution is normal. Loosely speaking, an estimator is consistent if for large samples the probability is close to one that the estimator will be near the population parameter being estimated. Explain why they are consistent. (Hint: what happens to the standard error of the sample mean as n gets larger and larger?)