Critical Thinking Suppose x has a distribution with m 72

Statistical Literacy What is the standard error of a sampling distribution?

Statistical Literacy What is the standard deviation of a sampling distribution called?

Statistical Literacy List two unbiased estimators and their corresponding parameters.

Statistical Literacy Describe how the variability of the distribution changes as the sample size increases

Statistical Literacy (a) If we have a distribution of x values that is more or less mound-shaped and somewhat symmetrical, what is the sample size needed to claim that the distribution of sample means from random samples of that size is approximately normal? b) If the original distribution of x values is known to be normal, do we need to make any restriction about sample size in order to claim that the distribution of sample means taken from random samples of a given size is normal?

Critical Thinking Suppose x has a distribution with m  72 and s  8. (a) If random samples of size n  16 are selected, can we say anything about the distribution of sample means? (b) If the original x distribution is normal, can we say anything about the distribution of random samples of size 16? Find P(68   73).

Critical Thinking Consider two \(\overline{x}\) distributions corresponding to the same x distribution. The first \(\overline{x}\) distribution is based on samples of size n = 100 and the second is based on samples of size n = 225. Which distribution has the smaller standard error? Explain.

Critical Thinking Consider an x distribution with standard deviation s  12. (a) If specifications for a research project require the standard error of the corresponding distribution to be 2, how large does the sample size need to be? (b) If specifications for a research project require the standard error of the corresponding distribution to be 1, how large does the sample size need to be?

Critical Thinking Suppose x has a distribution with m  15 and s  14. (a) If a random sample of size n  49 is drawn, find and P(15   17). (b) If a random sample of size n  64 is drawn, find and P(15   17). (c) Why should you expect the probability of part (b) to be higher than that of part (a)? (Hint: Consider the standard deviations in parts (a) and (b).)

Critical Thinking Suppose an x distribution has mean m  5. Consider two corresponding distributions, the first based on samples of size n  49 and the second based on samples of size n  81. (a) What is the value of the mean of each of the two distributions? (b) For which distribution is P(  6) smaller? Explain. (c) For which distribution is P(4 6) greater? Explain.

Coal: Automatic Loader Coal is carried from a mine in West Virginia to a power plant in New York in hopper cars on a long train. The automatic hopper car loader is set to put 75 tons of coal into each car. The actual weights of coal loaded into each car are normally distributed, with mean m  75 tons and standard deviation s  0.8 ton. (a) What is the probability that one car chosen at random will have less than 74.5 tons of coal? (b) What is the probability that 20 cars chosen at random will have a mean load weight of less than 74.5 tons of coal? (c) Interpretation: Suppose the weight of coal in one car was less than 74.5 tons. Would that fact make you suspect that the loader had slipped out of adjustment? Suppose the weight of coal in 20 cars selected at random had an average of less than 74.5 tons. Would that fact make you suspect that the loader had slipped out of adjustment? Why?

Vital Statistics: Heights of Men The heights of 18-year-old men are approximately normally distributed, with mean 68 inches and standard deviation 3 inches (based on information from Statistical Abstract of the United States, 112th Edition). (a) What is the probability that an 18-year-old man selected at random is between 67 and 69 inches tall? (b) If a random sample of nine 18-year-old men is selected, what is the probability that the mean height is between 67 and 69 inches? (c) Interpretation: Compare your answers to parts (a) and (b). Is the probability in part (b) much higher? Why would you expect this?

Medical: Blood Glucose Let x be a random variable that represents the level of glucose in the blood (milligrams per deciliter of blood) after a 12-hour fast. Assume that for people under 50 years old, x has a distribution that is approximately normal, with mean m  85 and estimated standard deviation s  25 (based on information from Diagnostic Tests with Nursing Applications, edited by S. Loeb, Springhouse). A test result x 40 is an indication of severe excess insulin, and medication is usually prescribed. (a) What is the probability that, on a single test, x 40? (b) Suppose a doctor uses the average for two tests taken about a week apart. What can we say about the probability distribution of Hint: See Theorem 7.1. What is the probability that 40? (c) Repeat part (b) for n  3 tests taken a week apart. (d) Repeat part (b) for n  5 tests taken a week apart. (e) Interpretation: Compare your answers to parts (a), (b), (c), and (d). Did the probabilities decrease as n increased? Explain what this might imply if you were a doctor or a nurse. If a patient had a test result of 40 based on five tests, explain why either you are looking at an extremely rare event or (more likely) the person has a case of excess insulin.

Medical: White Blood Cells Let x be a random variable that represents white blood cell count per cubic milliliter of whole blood. Assume that x has a distribution that is approximately normal, with mean m  7500 and estimated standard deviation s  1750 (see reference in Problem 13). A test result of x 3500 is an indication of leukopenia. This indicates bone marrow depression that may be the result of a viral infection. (a) What is the probability that, on a single test, x is less than 3500? (b) Suppose a doctor uses the average for two tests taken about a week apart. What can we say about the probability distribution of What is the probability of 3500? (c) Repeat part (b) for n  3 tests taken a week apart. (d) Interpretation: Compare your answers to parts (a), (b), and (c). How did the probabilities change as n increased? If a person had 3500 based on three tests, what conclusion would you draw as a doctor or a nurse?

Wildlife: Deer Let x be a random variable that represents the weights in kilograms (kg) of healthy adult female deer (does) in December in Mesa Verde National Park. Then x has a distribution that is approximately normal with mean \(\mu = 63.0\ kg\) and standard deviation \(\sigma = 7.1\ kg\) (Source: The Mule Deer of Mesa Verde National Park, by G. W. Mierau and J. L. Schmidt, Mesa Verde Museum Association). Suppose a doe that weighs less than 54 kg is considered undernourished. (a) What is the probability that a single doe captured (weighed and released) at random in December is undernourished? (b) If the park has about 2200 does, what number do you expect to be undernourished in December? (c) Interpretation: To estimate the health of the December doe population, park rangers use the rule that the average weight of n = 50 does should be more than 60 kg. If the average weight is less than 60 kg, it is thought that the entire population of does might be undernourished. What is the probability that the average weight \(\overline{x}\) for a random sample of 50 does is less than 60 kg (assume a healthy population)? (d) Interpretation: Compute the probability that \(\overline{x} < 64.2\ kg\) for 50 does (assume a healthy population). Suppose park rangers captured, weighed, and released 50 does in December, and the average weight was \(\overline{x} = 64.2\ kg\). Do you think the doe population is undernourished or not? Explain.

Focus Problem: Impulse Buying Let x represent the dollar amount spent on supermarket impulse buying in a 10-minute (unplanned) shopping interval. Based on a Denver Post article, the mean of the x distribution is about $20 and the estimated standard deviation is about $7. (a) Consider a random sample of n = 100 customers, each of whom has 10 minutes of unplanned shopping time in a supermarket. From the central limit theorem, what can you say about the probability distribution of \(\overline{x}\), the average amount spent by these customers due to impulse buying? What are the mean and standard deviation of the \(\overline{x}\) distribution? Is it necessary to make any assumption about the x distribution? Explain. (b) What is the probability that \(\overline{x}\) is between $18 and $22? (c) Let us assume that x has a distribution that is approximately normal. What is the probability that x is between $18 and $22? (d) Interpretation: In part (b), we used \(\overline{x}\), the average amount spent, computed for 100 customers. In part (c), we used x, the amount spent by only one customer. The answers to parts (b) and (c) are very different. Why would this happen? In this example, \(\overline{x}\) is a much more predictable or reliable statistic than x. Consider that almost all marketing strategies and sales pitches are designed for the average customer and not the individual customer. How does the central limit theorem tell us that the average customer is much more predictable than the individual customer?

Finance: Templeton Funds Templeton World is a mutual fund that invests in both U.S. and foreign markets. Let x be a random variable that represents the monthly percentage return for the Templeton World fund. Based on information from the Morningstar Guide to Mutual Funds (available in most libraries), x has mean \(\mu = 1.6%\) and standard deviation \(\sigma = 0.9%\). (a) Templeton World fund has over 250 stocks that combine together to give the overall monthly percentage return x. We can consider the monthly return of the stocks in the fund to be a sample from the population of monthly returns of all world stocks. Then we see that the overall monthly return x for Templeton World fund is itself an average return computed using all 250 stocks in the fund. Why would this indicate that x has an approximately normal distribution? Explain. Hint: See the discussion after Theorem 7.2. (b) After 6 months, what is the probability that the average monthly percentage return \(\overline{x}\) will be between 1% and 2%? Hint: See Theorem 7.1, and assume that x has a normal distribution as based on part (a). (c) After 2 years, what is the probability that \(\overline{x}\) will be between 1% and 2%? (d) Compare your answers to parts (b) and (c). Did the probability increase as n (number of months) increased? Why would this happen? (e) Interpretation: If after 2 years the average monthly percentage return \(\overline{x}\) was less than 1%, would that tend to shake your confidence in the statement that \(\mu = 1.6%\)? Might you suspect that \(\mu\) has slipped below 1.6%? Explain.

Finance: Dean Witter Funds Dean Witter European Growth is a mutual fund that specializes in stocks from the British Isles, continental Europe, and Scandinavia. The fund has over 100 stocks. Let x be a random variable that represents the monthly percentage return for this fund. Based on information from Morningstar (see Problem 17), x has mean m  1.4% and standard deviation s  0.8%. (a) Lets consider the monthly return of the stocks in the Dean Witter fund to be a sample from the population of monthly returns of all European stocks. Is it reasonable to assume that x (the average monthly return on the 100 stocks in the Dean Witter European Growth fund) has a distribution that is approximately normal? Explain. Hint: See Problem 17, part (a). (b) After 9 months, what is the probability that the average monthly percentage return will be between 1% and 2%? Hint: See Theorem 7.1 and the results of part (a). (c) After 18 months, what is the probability that the average monthly percentage return will be between 1% and 2%? (d) Compare your answers to parts (b) and (c). Did the probability increase as n (number of months) increased? Why would this happen? (e) Interpretation: If after 18 months the average monthly percentage return is more than 2%, would that tend to shake your confidence in the statement that m  1.4%? If this happened, do you think the European stock market might be heating up? Explain.

Expand Your Knowledge: Totals Instead of Averages Let x be a random variable that represents checkout time (time spent in the actual checkout process) in minutes in the express lane of a large grocery. Based on a consumer survey, the mean of the x distribution is about m  2.7 minutes, with standard deviation s  0.6 minute. Assume that the express lane always has customers waiting to be checked out and that the distribution of x values is more or less symmetrical and mound-shaped. What is the probability that the total checkout time for the next 30 customers is less than 90 minutes? Let us solve this problem in steps. (a) Let xi (for i  1, 2, 3, . . . , 30) represent the checkout time for each customer. For example, x1 is the checkout time for the first customer, x2 is the checkout time for the second customer, and so forth. Each xi has mean m  2.7 minutes and standard deviation s  0.6 minute. Let w  x1 x2 . . . x30. Explain why the problem is asking us to compute the probability that w is less than 90. (b) Use a little algebra and explain why w 90 is mathematically equivalent to w/30 3. Since w is the total of the 30 x values, then w/30  Therefore, the statement 3 is equivalent to the statement w 90. From this we conclude that the probabilities P( 3) and P(w 90) are equal. (c) What does the central limit theorem say about the probability distribution of Is it approximately normal? What are the mean and standard deviation of the distribution? (d) Use the result of part (c) to compute P( 3). What does this result tell you about P(w 90)?

Totals Instead of Averages: Airplane Takeoff Time The taxi and takeoff time for commercial jets is a random variable x with a mean of 8.5 minutes and a standard deviation of 2.5 minutes. Assume that the distribution of taxi and takeoff times is approximately normal. You may assume that the jets are lined up on a runway so that one taxies and takes off immediately after the other, and that they take off one at a time on a given runway. What is the probability that for 36 jets on a given runway, total taxi and takeoff time will be (a) less than 320 minutes? (b) more than 275 minutes? (c) between 275 and 320 minutes? Hint: See Problem 19.

Totals Instead of Averages: Escape Dunes It’s true—sand dunes in Colorado rival sand dunes of the Great Sahara Desert! The highest dunes at Great Sand Dunes National Monument can exceed the highest dunes in the Great Sahara, extending over 700 feet in height. However, like all sand dunes, they tend to move around in the wind. This can cause a bit of trouble for temporary structures located near the “escaping” dunes. Roads, parking lots, campgrounds, small buildings, trees, and other vegetation are destroyed when a sand dune moves in and takes over. Such dunes are called “escape dunes” in the sense that they move out of the main body of sand dunes and, by the force of nature (prevailing winds), take over whatever space they choose to occupy. In most cases, dune movement does not occur quickly. An escape dune can take years to relocate itself. Just how fast does an escape dune move? Let x be a random variable representing movement (in feet per year) of such sand dunes (measured from the crest of the dune). Let us assume that x has a normal distribution with \(\mu = 17\) feet per year and \(\sigma = 3.3\) feet per year. (For more information, see Hydrologic, Geologic, and Biologic Research at Great Sand Dunes National Monument and Vicinity, Colorado, proceedings of the National Park Service Research Symposium.) Under the influence of prevailing wind patterns, what is the probability that (a) an escape dune will move a total distance of more than 90 feet in 5 years? (b) an escape dune will move a total distance of less than 80 feet in 5 years? (c) an escape dune will move a total distance of between 80 and 90 feet in 5 years? Hint: See Problem 19 and Theorem 7.1.