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Upper level probability

### An analysis of variance of the yields of five different varieties of wheat, observed on one plot each at each of six different locations. The data from this randomized block design are listed here: Varieties, Location 1, Location 2, Location 3, Location 4, Location 5, Location 6 A, 35.3, 31.0, 32.7, 36.8, 37.2, 33.1 B, 30.7, 32.2, 31.4, 31.7, 35.0, 32.7 C, 38.2, 33.4, 33.6, 37.1, 37.3, 38.2 D, 34.9, 36.1, 35.2, 38.3, 40.2, 36.0 E, 32.4, 28.9, 29.2, 30.7, 33.9, 32.1 a. Use the appropriate nonparametric test to determine whether the data provide sufficient evidence to indicate a difference in the yields for the five different varieties of wheat. Test using αα=.05. b. How do the analysis of variance F test compare with the test in part a? Explain.

Upper level probability

### 6% of male are color-blind. Suggest a male groub is randomly selected and tested, find the probability that the first color-blind will be found among the first 5 men tested.

Upper level probability

### The doctor says there is a 75% probability that you have a kidney disorder or are diabetic. What is the probability that you both have a kidney and are diabetic if there is 40% chance kidney disorder and a 50% chance diabetic.

Upper level probability

### A balanced experimental design has a sample size of $$n=11$$ observations at each of $$k=3$$ factor levels. The sample averages are $$\overline{x}_{1⋅}=48.05,\overline{x}_{2⋅}=44.74, and\ \overline{x}_{3⋅}=49.11, and\ MSE=4.96$$. (a) Calculate pairwise confidence intervals for the factor level means with an overall confidence level of 95%. (b) Make a diagram showing which factor level means are known to be different and which ones are indistinguishable. (c) What additional sampling would you recommend to reduce the lengths of the pairwise confidence intervals to no more than 2.0 ?

Upper level probability

### A study of about n=1000 individuals in the United States during September 21-22, 2001, revealed that 43% of the respondents indicated that they were less willing to fly following the events of September 11, 2001. a. Is this an observational study or a designed experiment? b. What problems might or could have occurred because of the sensitive nature of the subject? What kinds of biases might have occurred?

Upper level probability

### The analysis of variance table for a $$3 \times 4$$ factorial experiment, with factor A at three levels and factor B at four levels, and with two observations per treatment is shown here: Source, df, SS, MS, F A, 2, 5.3 B, 3, 9.1 AB, 6, Error, 12, 24.5 Total, 23, 43.7 a. Fill in the missing items in the table. b. Do the data provide sufficient evidence to indicate that factors A and B interact? Test using $$\alpha \alpha=.05$$. What are the practical implications of your answer? c. Do the data provide sufficient evidence to indicate that factors A and B affect the response variable x? Explain.

Upper level probability

### Toss a fair die four times. What is the probability that all tosses produce different outcomes?

Upper level probability

### To evaluate $$\int \tan^3 (\pi x)dx$$

Upper level probability

### There are three women and four men in a group of seven people. If three people are selected from the total of seven, find the following: i)What are the total possible outcomes for this selection? ii)How many ways can two women and one man be selected? iii)What is the probability of selecting two women and one man?

Upper level probability

### Derive $$100(1-\alpha)$$ percent lower and upper confidence intervals for p, when the data consist of the values of n independent Bernoulli random variables with parameter p.

Upper level probability

### The paired comparisons option in MINITAB generated the output provided here. What do these results tell you about the differences in the population means? Does this confirm your conclusion of the analysis-of-variance? MINITAB output Tukey's 95% Simultaneous Confidence Intervals All Pairwise Comparisons among Levels of Method Individual confidence level=97.94% Method = 1 subtracted from: Method 2: Lower -0.0001777, Center -0.0008400, Upper -0.0002423 Method 3: Lower -0.0001777, Center 0.0004200, Upper 0.0010177 Method = 2 subtracted from: Method 3: Lower 0.0006623, Center 0.0012600, Upper 0.0018577

Upper level probability

### Suppose we have two independent random variables x1 and x2 with respective population means μ1 and μ2. Let us say that we use sample data to construct two 80% confidence intervals. Confidence Interval amp; Confidence Level A1 lt;μ1 lt;B1 amp;0.80 ​ A2 lt;μ2 lt;B2 amp;0.80 Now, what is the probability that both intervals hold at the same time? Use methods to show that P(A1<μ1

Upper level probability

### An analysis of variance can be used to compare: A. Two treatments with categorical data B. Three treatments with continuous data C. Three treatments with categorical data D. None of the above

Upper level probability

### If the p-value in the ANOVA table for a one-way analysis is larger than 10%, then: A. The typical follow-up confidence intervals (with confidence level 95%) for the pairwise comparisons will all contain zero. B. Some of the typical follow-up confidence intervals (with confidence level 95%) for the pairwise comparisons may not contain zero. C. None of the typical follow-up confidence intervals (with confidence level 95%) for the pairwise comparisons can contain zero. D. None of the above.

Upper level probability

### an urn contains 1 white, 1 green, and 2 red balls. draw 3 balls with replacement. find probability we did not see all the colours

Upper level probability

### An analysis of reading test scores of students at a rural Texas school district was carried out in Current Issues in Education (Jan. 2014). Students were classified as attending elementary, middle, or high school and whether they passed a reading comprehension test. Toe data for the sample of 1,012 students are summarized in the accompanying table. Does the passing rate on the reading comprehension test in Texas differ for elementary, middle, and high school students? Use $$\alpha=10$$. $$\begin{array}{|l|c|c|c|} \hline& \text {Elementary School} & \text {Middle School} & \text {High School} \\ \hline \text {Number Passing} &372 &418 & 143 \\ \hline \text {Number Failing} &44 &25 & 10 \\ \hline \text {Totals } &416 & 443 & 153\\ \hline \end{array}$$

Upper level probability

### The probability of a baby being born a boy is 0.512. Consider the problem of finding the probability of exactly 7 boys in 11 births. Solve that problem using normal approximation to the binomial.

Upper level probability

### In Major League baseball, do Left-Handed hitters and Right-Handed hitters tend to have different batting averages? We will study this question by taking random samples of m=45 seasonal batting averages from left-handed batters and n=53 seasonal batting averages from right-handed batters. The mean batting averages were x= .256 for the left-handers and y=.254 for the right-handers. It is recognized that the true standard deviations of batting averages are σx = .0324 for left-handed batters and σy = .0298 for right-handed batters. The true (unknown) mean batting average for left-handers = μx , while the true unknown mean batting average for right-handers = μy . We would like to examine μx − μy . a) What is the standard deviation of the distribution of x? b) What is the standard deviation of the distribution of x - y ? c) Create a 98% confidence interval for μx − μy ? leftparen1.gif , rightparen1.gif d) What is the length of the confidence interval in part c) ?

Upper level probability