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| m102805 | There are several extensions of linear regression that apply to exponential growth and power law models. Problems 22-25 will outline some of these extensions. First of all, recall that a variable grows linearly over time if it adds a fi xed increment during each equal time period. Exponential growth occurs when a variable is multiplied by a fi xed number during each time period. This means that exponential growth increases by a fixed multiple or percentage of the previous amount. College algebra can be used to show that if a variable grows exponentially, then its logarithm grows linearly. The exponential growth model is y = αβx, where a and b are fixed constants to be estimated from data.
How do we know when we are dealing with exponential growth, and how can we estimate α and β? Please read on. Populations of living things such as bacteria, locusts, sh, panda bears, and so on, tend to grow (or decline) exponentially. However, these populations can be restricted by outside limitations such as food, space, pollution, disease, hunting, and so on. Suppose we have data pairs (x, y) for which there is reason to believe the scatter plot is not linear, but rather exponential, as described above. This means the increase in y values begins rather slowly but then seems to explode. Note: For exponential growth models, we assume all y > 0.
Consider the following data, where x = time in hours and y = number of bacteria in a laboratory culture at the end of x hours.
(a) Look at the |
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| m102806 | Thirty years ago, the Bureau of Justice Statistics reported in Profile of Jail Inmates that the median educational attainment of jail inmates was 10.2 years. Ten current inmates are randomly selected and found to have the following educational attainments, in years.
Assume that educational attainments of current jail inmates have a symmetric, non-normal distribution. At the 10% significance level, do the data provide sufficient evidence to conclude that this year s median educational attainment has changed from what it was 30 years ago?
a. Use the t-test.
b. Use the Wilcoxon signed-rank test.
c. If this year s median educational attainment has in fact changed from what it was 30 years ago, how do you explain the discrepancy between the two tests? |
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| m102807 | Thirty-six percent of all Americans drink bottled water more than once a week (Natural Resources Defense Council, December 4, 2015). Suppose you have been hired by the Natural Resources Defense Council to investigate bottled water consumption in St. Paul. You plan to select a sample of St. Paulites to estimate the proportion who drink bottled water more than once a week. Assume the population proportion of St. Paulites who drink bottled water more than once a week is .36, the same as the overall proportion of Americans who drink bottled water more than once a week.
a. Suppose you select a sample of 540 St. Paulites. Show the sampling distribution of p.
b. Based upon a sample of 540 St. Paulites, what is the probability that the sample proportion will be within .04 of the population proportion?
c. Suppose you select a sample of 200 St. Paulites. Show the sampling distribution of p.
d. Based upon the smaller sample of only 200 St. Paulites, what is the probability that the sample proportion will be within .04 of the population proportion?
e. As measured by the increase in probability, how much do you gain in precision by taking the larger sample in parts (a) and (b) rather than the smaller sample in parts (c) and (d)? |
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| m102808 | This cumulative review problem uses material from Chapters 3, 5, and 10. Recall that the Poisson distribution deals with rare events. Death from the kick of a horse is a rare event, even in the Prussian army. The following data are a classic example of a Poisson application to rare events.
A reproduction of the original data can be found in C. P. Winsor, Human Biology, Vol. 19, pp. 154-161. The data represent the number of deaths from the kick of a horse per army corps per year for 10 Prussian army corps for 20 years (1875-1894). Let x represent the number of deaths and f the frequency of x deaths.
(a) First, we t the data to a Poisson distribution (see Section 5.4).
Poission distribution:
Where λ ≈ x̅ (sample mean of x values) From our study of weighted averages (see section 3.1),
Verify that x̅ ≈ 0.61 For the category 3 or more, use 3.
(b) Now we have
1, 2, 3. p
Find P(0), P(1), P(2), and P(3 ≤ x). Round to three places after the decimal.
(c) The total number of observations is ∑f = 200.
For a given x, the expected frequency of x deaths is 200P(x). The following table gives the observed frequencies O and the expected frequencies E = 200P(x).
(d) State the null and alternate hypotheses for a chi-square goodness-of- t test. Set the level of signifi cance to be a = 0.01. Find the P-value for a goodness-of- t test. Interpret your conclusion in the context of this application. Is there reason to believe that the Poisson distribution fits the raw data pro |
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| m102809 | This exercise requires that you have first done Exercise 3.199. In Example 3.12 on page 109, we found that, considering the five starting players on Team II a sample of all male starting college basketball players, the mean and standard deviation of the heights are 75 inches and 6.2 inches, respectively. Explain why, numerically, the sample mean of 75 inches is the same as the population mean found in Exercise 3.199(a) but that the sample standard deviation of 6.2 inches differs from the population standard deviation found in Exercise 3.199(b). |
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| m102810 | This is a technique to break down the variation of a random variable into useful components (called stratum) in order to decrease experimental variation and increase accuracy of results. It has been found that a more accurate estimate of population mean m can often be obtained by taking measurements from naturally occurring subpopulations and combining the results using weighted averages. For example, suppose an accurate estimate of the mean weight of sixth grade students is desired for a large school system. Suppose (for cost reasons) we can only take a random sample of m = 100 students, Instead of taking a simple random sample of 100 students from the entire population of all sixth grade students, we use stratifi ed sampling as follows. The school system under study consists three large schools. School A has N1 = 310 sixth grade students, School B has N2 = 420 sixth grade students, and School C has N3 = 516 sixth grade students. This is a total population of 1246 sixth grade students in our study and we have strata consisting of the 3 schools. A preliminary study in each school with relatively small sample size has given estimates for the sample standard deviation s of sixth grade student weights in each school. These are shown in the following table.
How many students should we randomly choose from each school for a best estimate m for the population mean weight? A lot of mathematics goes into the answer. Fortunately, Bill Williams of Bell Laboratories wrote a book calle |
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| m102811 | This problem continues the Focus Problem. The solution involves applying several basic probability rules and a little algebra to solve an equation.
(a) If the polygraph of Problem 27 indicated that 30% of the questions were answered with lies, what would you estimate for the actual percentage of lies in the answers? Let B = event detector indicates a lie. We are given P (B) = 0.30. Let A = event person is lying, so Ac = event person is not lying. Then
P (B) = P (A and B) + P (Ac and B)
P (B) = P (A) P (B | A) + P (Ac) P (B | Ac)
Replacing P (Ac) by 1 - P (A) gives
P (B) = P (A) ( P (B | A) + [1 - P (A)] ( P (B | Ac)
Substitute known values for P (B), P (B | A), and P (B | Ac) into the last equation and solve for P (A).
(b) If the polygraph indicated that 70% of the questions were answered with lies, what would you estimate for the actual percentage of lies? |
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| m102812 | This problem is based on information taken from Life in America s Fifty States by G. S. Thomas. A random sample of n1 = 153 people ages 16 to 19 was taken from the island of Oahu, Hawaii, and 12 were found to be high school dropouts. Another random sample of n2 = 128 people ages 16 to 19 was taken from Sweetwater County, Wyoming, and 7 were found to be high school dropouts. Do these data indicate that the population proportion of high school dropouts on Oahu is different (either way) from that of Sweetwater County? Use a 1% level of signifi cance.
(a) What is the level of signi ficance? State the null and alternate hypotheses.
(b) What sampling distribution will you use? What assumptions are you making? Compute the sample test statistic and corresponding z or t value as appropriate.
(c) Find (or estimate) the P-value. Sketch the sampling distribution and show the area corresponding to the P-value.
(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically signifi cant at level a?
(e) Interpret your conclusion in the context of the application.
For degrees of freedom d.f. not in the Student s t table, use the closest d.f. that is smaller. In some situations, this choice of d.f. may increase the P-value a small amount and therefore produce a slightly more "conservative" answer.
Answers may vary due to rounding. |
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| m102813 | This problem shows you how to make a better blend of almost anything.
Let x1, x2, ... xn be independent random variables with respective variances (21, (22 , ... , (2n.
Let c1, c2, .........., cn be constant weights such that 0 ( ci ( 1 and c1 + c2 + ... + cn = 1. The linear combination w = c1x1 + c2 x2 + ... + cnxn is a random variable with variance
(a) Two types of epoxy resin are used to make a new blend of superglue. Both resins have about the same mean breaking strength and act independently. The question is how to blend the resins (with the hardener) to get the most consistent breaking strength. Why is this important, and why would this require minimal (2W?
We don t want some bonds to be really strong while others are very weak, resulting in inconsistent bonding.
Let x1 and x2 be random variables representing breaking strength (lb) of each resin under uniform testing conditions. If (1 = 8 lb (2 = 12 lb, show why a blend of about 69% resin 1 and 31% resin 2 will result in a superglue with smallest (W2 and most consistent bond strength.
(b) Use c1 = 0.69 and c2 = 0.31 to compute (w and show that (w is less than both s1 ands2. The dictionary meaning of the word synergetic is "working together or cooperating for a better overall effect." Write a brief explanation of how the blend w = c1x1 + c2 x2 has a synergetic effect for the purpose of reducing variance? |
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| m102814 | This problem will be referred to in the study of control charts (Section 6.1). In the binomial probability distribution, let the number of trials be n = 3, and let the probability of success be p = 0.0228. Use a calculator to compute
(a) The probability of two successes.
(b) The probability of three successes.
(c) The probability of two or three successes. |
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| m102815 | Three events A, B, and C are said to be independent if
P(A & B) = P(A) · P(B),
P(A & C) = P(A) · P(C),
P(B & C) = P(B) · P(C), and
P(A & B & C) = P(A) · P(B) · P(C).
What is required for four events to be independent? Explain your definition in words. |
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| m102816 | Three grades of jet fuel are blended to be used on long commercial flights. Each grade cost about the same and has about the same ignition or burn rate (gal/min) in a jet engine. The question is how to make the blend for the most consistent ignition rate. Assume the ignition rates of the three grades of fuel are statistically independent.
(a) Let x1, x2, x3 be random variables representing ignition rate (gal/min) for the three grades of fuel. After extensive testing (flying under different conditions) the standard deviations are found to be (1 = 7, (2 = 12 and (3 = 15. Explain why a blend of approximately 64% grade 1 fuel with 22% grade 2 fuel and 14% grade 3 fuel would give the most consistent ignition rate.
(b) Use c1 = 0.64, c2 = 0.22, and c3 = 0.14 to compute (w and show that (w is less than each of (1, (2 and (3. Would you say the blended jet fuel has a more consistent ignition rate than each separate fuel? Explain. |
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| m102817 | Three local districts are "swing" districts for an upcoming election on a contentious political issue. A survey will be conducted in which voters will be asked to rate their opinion regarding this issue on a scale of 0 (strongly oppose) to 10 (strongly support). A small preliminary random sample from each district was used to estimate the sample standard deviation s of responses for the district. The following table shows the number of voters N in each district and the sample standard deviation s of strength of support in each district.
District 1 __________ District 2 __________ District 3
N1 = 1525 .............. N2 = 917 .............. N3 = 2890
s1 = 2.2 .................. s2 = 1.4 .................. s3 = 3.3
We have a total population of 5332 voters and 3 strata (districts). The group doing the survey has enough funding to obtain a random sample of m = 250 total responses from all the districts.
(a) Compute the size of the random samples n1, n2, n3 to taken from each district. Round each sample size to the nearest whole number and make sure they add up to m = 250.
(b) Suppose you took the appropriate random samples from each district and you got the following average political support measures (scale 0 to 10): x̅1 = 6.2, x̅2 = 3.1 x̅3 = 8.5. Compute your best estimate for the population mean m (scale 0 to 10) of voter support for the issue from the total population of these three districts. |
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| m102818 | To better understand eating patterns that might contribute to obesity, a researcher measures the average number of calories (per meal) consumed by shift workers (morning, afternoon, night) during two seasons (summer and winter). The hypothetical results are given in the following table.
(a) Complete the F table and make a decision to retain or reject the null hypothesis for each hypothesis test.
(b) Explain why post hoc tests are not necessary. |
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| m102819 | To compare two elementary schools regarding teaching of reading skills, 12 sets of identical twins were used. In each case, one child was selected at random and sent to school A, and his or her twin was sent to school B. Near the end of fifth grade, an achievement test was given to each child. The results follow:
Use a 0.05 level of signifi cance to test the hypothesis that the two schools have the same effectiveness in teaching reading skills against the alternate hypothesis that the schools are not equally effective.
(a) What is the level of signi ficance? State the null and alternate hypotheses.
(b) Compute the sample test statistic. What is the sampling distribution?
(c) Find the P-value of the sample test statistic.
(d) Conclude the test.
(e) Interpret the conclusion in the context of the application. |
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| m102820 | To demonstrate flavor aversion learning (that is, learning to dislike a flavor that is associated with becoming sick), researchers gave one group of laboratory rats an injection of lithium chloride immediately following consumption of saccharin-flavored water. Lithium chloride makes rats feel sick. A control group was not made sick after drinking the flavored water. The next day, both groups were allowed to drink saccharin-flavored water. The amounts consumed (in milliliters) for both groups during this test are given below.
(a) Test whether or not consumption of saccharin-flavored water differed between groups using a .05 level of significance. State the value of the test statistic and the decision to retain or reject the null hypothesis.
(b) Compute effect size using eta-squared (η2). |
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| m102821 | To test the relationship between sex and responsiveness to feedback, a group of men and women were given a vignette describing a person of the opposite sex who gave negative, positive, or neutral feedback about relationship advice. Participants rated how positively they felt about the individual described in the vignette, with higher ratings indicating more positive feelings.
(a) Complete the F table and make a decision to retain or reject the null hypothesis for each hypothesis test.
(b) Based on the results you obtained, what is the next step? |
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| m102822 | To test whether animal subjects consume the same amounts of sweet-tasting solutions, a researcher has 10 subjects consume one of three sweet-tasting solutions: sucrose, saccharin, or Polycose. The amount consumed (in milliliters) of each solution is given in the table.
(a) Complete the F table.
(b) Compute Tukey s HSD post hoc test and interpret the results. |
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| m102823 | To test whether arousal or stress levels increase as the difficulty of a task increases, 10 participants were asked to complete an easy, typical, or difficult task. Their galvanic skin response (GSR) was recorded. A GSR measures the electrical signals of the skin in units called micro Siemens (μS), with higher signals indicating greater arousal or stress. The data for each task are given in the table.
(a) Complete the F table.
(b) Compute Fisher s LSD post hoc test and interpret the results. |
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| m102824 | To test whether students in a higher grade level would be less disruptive in class, a school psychologist recorded the number of documented interruptions during one day of classes from nine local high schools. The sample consisted of nine freshman, sophomore, junior, and senior high school classes. The data for each high school class are given in the table.
(a) Complete the F table.
(b) Is it necessary to compute a post hoc test? Explain. |
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