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| m102825 | Tri-County Bank is a small independent bank in central Wyoming. This is a rural bank that makes loans on items as small as horses and pickup trucks to items as large as ranch land. Total monthly loan requests are used by bank of ficials as an indicator of economic business conditions in this rural community. The mean monthly loan request for the past several years has been 615.1 (in thousands of dollars) with a standard deviation of 11.2 (in thousands of dollars). The distribution of loan requests is approximately mound-shaped and symmetric.
(a) For 12 months, the following monthly loan requests (in thousands of dollars) were made to Tri-County Bank:
Make a control chart for the total monthly loan requests, and plot the preceding data on the control chart. Interpretation From the control chart, would you say the local business economy is heating up or cooling down? Explain your answer by referring to any trend you may see on the control chart. Identify all out-of-control signals by type (I, II, or III).
(b) For another 12-month period, the following monthly loan requests (in thousands of dollars) were made to Tri-County Bank:
Make a control chart for the total monthly loan requests, and plot the preceding data on the control chart. Interpretation From the control chart, would you say the local business economy is heating up, cooling down, or about normal? Explain your answer by referring to the control chart. Identify all out-of-control signals by type (I, II, or III) |
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| m102826 | Tung and Deng (2007) studied how emoticons (computerized facial expressions) and sex influenced intrinsic motivation to complete a computer task. Emoticons were used as feedback for responses during the task. These faces were either static (presented still for the duration of feedback) or dynamic (presented as a neutral face and changed to the expression corresponding to the feedback). The researchers conducted a 2 × 2 between-subjects ANOVA, with emoticon style (static, dynamic) and sex as the factors. The F table gives the data reported in the study.
(a) Is a significant main effect or interaction evident in this study and, if so, for which factor?
(b) Are post hoc tests required? Explain. |
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| m102827 | Tunney (2006) was interested in various probability relationships. Specifically, he was interested in how the presentation of probabilities (either as a probability or as a frequency) influenced participant choices. The following is an example of how he presented bets for frequency and probability. Both statements say the same thing for each bet but in different ways.
Based on this table,
(a) What is the mathematical expectation for winning the Pbet,
(b) What is the mathematical expectation for winning the £-bet? Based on the mathematical expectation of making P-bets and £-bets,
(c) If you had to make one bet, which one would you make? Explain. |
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| m102828 | Turbid water is muddy or cloudy water. Sunlight is necessary for most life forms; thus turbid water is considered a threat to wetland ecosystems. Passive filtration systems are commonly used to reduce turbidity in wetlands. Suspended solids are measured in mg/L. Is there a relation between input and output turbidity for a passive fi ltration system and, if so, is it statistically signifi cant? At a wetlands environment in Illinois, the inlet and outlet turbidity of a passive filtration system have been measured. A random sample of measurements follow.
(i) Rank-order the inlet readings using 1 as the largest data value. Also rank-order the outlet readings using 1 as the largest data value. Then construct a table of ranks to be used for a Spearman rank correlation test.
(ii) Use a 1% level of signi ficance to test the claim that there is a monotone relation ship (either way) between the ranks of the inlet readings and outlet readings.
(a) What is the level of signi ficance? State the null and alternate hypotheses.
(b) Compute the sample test statistic.
(c) Find or estimate 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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| m102829 | Twenty-two fourth-grade children were randomly divided into two groups. Group A was taught spelling by a phonetic method. Group B was taught spelling by a memorization method. At the end of the fourth grade, all children were given a standard spelling exam.
The scores are as follows.
Use a 1% level of signifi cance to test the claim that there is no difference in the test score distributions based on instruction method.
(a) What is the level of signifi cance? State the null and alternate hypotheses.
(b) Compute the sample test statistic. What is the sampling distribution?
What conditions are necessary to use this 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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| m102830 | Two balanced dice are rolled. Refer to Fig. 4.1 on page 159 and determine the probability that the sum of the dice is
In Figure 4.1
a. 6.
b. Even.
c. 7 or 11.
d. 2, 3, or 12. |
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| m102831 | Two balanced dice are thrown, one red and one black. What is the probability that the red die comes up 1, given that the
a. Black die comes up 3?
b. Sum of the dice is 4?
c. Sum of the dice is 9? |
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| m102832 | Two cards are drawn at random from an ordinary deck of 52 cards. Determine the probability that both cards are aces if
a. The first card is replaced before the second card is drawn.
b. The first card is not replaced before the second card is drawn. |
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| m102833 | Two expert French chefs judged chocolate mousse made by students in a Paris cooking school. Each chef ranked the best chocolate mousse as 1.
Use a 0.10 level of signifi cance to test the claim that there is a monotone relation (either way) between ranks given by Chef Pierre and by Chef André.
(a) State the test used.
(b) Give a. State the null and alternate hypotheses.
(c) Find the sample test statistic.
(d) For the sign test, rank-sum test, and Spearman correlation coef ficient test, fi nd the P-value of the sample test statistic. For the runs test of randomness, find the critical values from Table 10 of Appendix II.
(e) Conclude the test and interpret the results in the context of the application |
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| m102834 | Two plots at Rothamsted Experimental Station (see reference in Problem 5) were studied for production of wheat straw. For a random sample of years, the annual wheat straw production (in pounds) from one plot was as follows:
Use a calculator to verify that, for the preceding data, s2 ( 0.318.
Another random sample of years for a second plot gave the following annual wheat straw production (in pounds):
Use a calculator to verify that, for these data, s2 ( 1.078.
Test the claim that there is a difference (either way) in the population variance of wheat straw production for these two plots. Use a 5% level of signifi cance?
Please provide the following information.
(a) What is the level of signi ficance? State the null and alternate hypotheses.
(b) Find the value of the sample F statistic. What are the degrees of freedom?
What assumptions are you making about the original distribution?
(c) Find or estimate the P-value of the sample test statistic.
(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis?
(e) Interpret your conclusion in the context of the application. |
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| m102835 | Two processes for manufacturing 60-watt light bulbs are under study. In both cases, the life (in hours) of the bulb before it burns out is being examined. A random sample of 18 light bulbs manufactured using the old process showed the sample variance of lifetimes to be s2 = 51.87. Another random sample of 16 light bulbs manufactured using the new process showed the sample variance of the lifetimes to be s2 = 135.24. Use a 5% level of significance to test the claim that the population variance of lifetimes for the new manufacturing process is larger than that of the old process. |
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| m102836 | Two processes for manufacturing large roller bearings are under study. In both cases, the diameters (in centimeters) are being examined. A random sample of 21 roller bearings from the old manufacturing process showed the sample variance of diameters to be s2 = 0.235. Another random sample of 26 roller bearings from the new manufacturing process showed the sample variance of their diameters to be s2 = 0.128. Use a 5% level of signi ficance to test the claim that there is a difference (either way) in the population variances between the old and new manufacturing processes. |
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| m102837 | Umbilical cord blood analysis immediately after delivery is one way to measure the health of an infant after birth. Researchers G. Natalucci et al. used it as a predictor of brain maturation of preterm infants in the article "Functional Brain Maturation Assessed During Early Life Correlates with Anatomical Brain Maturation at Term-Equivalent Age in Preterm Infants" (Pediatric Research, Vol. 74. No. 1, pp. 68-74). Based on this study, we will assume that, for preterm infants, the pH level of the arterial cord (one vessel in the umbilical cord) is normally distributed with mean 7.32 and standard deviation 0.1. Find the percentage of preterm infants who have arterial cord pH levels
a. Between 7.0 and 7.5.
b. Over 7.4. |
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| m102838 | Under the condition that both populations have equal standard deviations (σ1 - σ2), we can pool the standard deviations and use a Student s t distribution with degrees of freedom d.f. - n1 + n2 - 2 to fi nd the margin of error of a c confi dence interval for µ1 - µ2. This technique demonstrates another commonly used method of computing confi dence intervals for µ1 - µ2.
PROCEDURE
How to find a confidence interval for µ1 - µ2 when σ1 - σ2
Requirements
Consider two independent random samples, where
x̅1 and x̅2 are sample means from populations 1 and 2
s1 and s2 are sample standard deviations from populations 1 and 2
n1 and n2 are sample sizes from populations 1 and 2
If you can assume that both population distributions 1 and 2 are normal or at least mound-shaped and symmetric, then any sample sizes n1 and n2 will work.
If you cannot assume this, then use sample sizes n1 ≥ 30 and n2 ≥ 30.
Confidence interval for µ1 - µ2 when σ1 - σ2
Where
(pooled standard deviation)
c = confidence level (0 < c < 1)
tc - critical value for confi dence level c and degrees of freedom
d.f. = n1 + n2 - 2
With statistical software, select pooled variance or equal variance options.
(a) There are many situations in which we want to compare means from populations having standard deviations that are equal. The pooled standard deviation method applies even if the standard deviations are known to be only approximately equal. Consider Problem 23 regarding weights of gray |
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| m102839 | Under what circumstances is using a normal probability plot to assess the normality of a variable usually better than using a histogram, stem-and-leaf diagram, or dot-plot? |
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| m102840 | Under what three conditions are repeated trials of an experiment called Bernoulli trials? |
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| m102841 | "Unknown cultural affi liations and loss of identity at high elevations." These words are used to propose the hypothesis that archaeological sites tend to lose their identity as altitude extremes are reached. This idea is based on the notion that prehistoric people tended not to take trade wares to temporary settings and/or isolated areas (Source: Prehistoric New Mexico: Background for Survey, by D. E. Stuart and R. P. Gauthier, University of New Mexico Press). As elevation zones of prehistoric people (in what is now the state of New Mexico) increased, there seemed to be a loss of artifact identifi cation. Consider the following information.
Let p1 be the population proportion of unidentifi ed archaeological artifacts at the elevation zone 7000-7500 feet in the given archaeological area. Let p2 be the population proportion of unidenti fied archaeological artifacts at the elevation zone 5000-5500 feet in the given archaeological area.
(a) Check Requirements Can a normal distribution be used to approximate the 1 - 2 distribution? Explain.
(b) Find a 99% con fidence interval for p1 - p2.
(c) Interpretation Explain the meaning of the confi dence interval in the context of this problem. Does the confi dence interval contain all positive numbers? all negative numbers? both positive and negative numbers? What does this tell you (at the 99% con fidence level) about the comparison of the population proportion of unidenti fied artifacts at high elevations (7000-7500 feet) wi |
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| m102842 | USA Today reported that about 20% of all people in the United States are illiterate. Suppose you interview seven people at random off a city street.
(a) Make a histogram showing the probability distribution of the number of illiterate people out of the seven people in the sample.
(b) Find the mean and standard deviation of this probability distribution. Find the expected number of people in this sample who are illiterate.
(c) How many people would you need to interview to be 98% sure that at least seven of these people can read and write (are not illiterate)? |
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| m102843 | USA Today reported that for all airlines, the number of lost bags was
May: 6.02 per 1000 passengers December: 12.78 per 1000 passengers
Note: A passenger could lose more than one bag.
(a) Let r = number of bags lost per 1000 passengers in May. Explain why the Poisson distribution would be a good choice for the random variable r. What is ( to the nearest tenth?
(b) In the month of May, what is the probability that out of 1000 passengers, no bags are lost? that 3 or more bags are lost? that 6 or more bags are lost?
(c) In the month of December, what is the probability that out of 1000 passengers, no bags are lost? that 6 or more bags are lost? that 12 or more bags are lost? (Round l to the nearest whole number.) |
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| m102844 | USA Today reported that Park eld, California, is dubbed the world s earthquake capital because it sits on top of the notorious San Andreas fault. Since 1857, Park eld has had a major earthquake on the average of once every 22 years.
(a) Explain why a Poisson probability distribution would be a good choice for r = number of earthquakes in a given time interval.
(b) Compute the probability of at least one major earthquake in the next 22 years. Round l to the nearest hundredth, and use a calculator.
(c) Compute the probability that there will be no major earthquake in the next 22 years. Round l to the nearest hundredth, and use a calculator.
(d) Compute the probability of at least one major earthquake in the next 50 years. Round l to the nearest hundredth, and use a calculator.
(e) Compute the probability of no major earthquakes in the next 50 years. Round ( to the nearest hundredth, and use a calculator. |
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