# Recent questions in Upper level probability

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

### 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 α=.10. ElementaryMiddleHigh SchoolSchoolSchool Number Passing372418143 Number Failing442510 Totals416443153

Upper level probability

### There are 2 Son Ss OF Orange juice, 3 bottles of guava juice and 2 bottles of apple yuice in a bow. If 2 bottles of juice are chosen at random from the box, find the probability of choosing, 2 borties of juice of different Bavours.

Upper level probability

### The probability that event A occurs is 0,3, the probability that event B does not occur is 0,4. What is the probability that events A and B occur simultaneously if $$\displaystyle{P}{\left({A}∪{B}\right)}={0},{7}$$?

Upper level probability

### Let A represent having soup and let B represent having salad for lunch. Which statement is true? Having soup and salad for lunch are independent events because $$\displaystyle{P}{\left({A}∣{B}\right)}={P}{\left({A}\right)}{\quad\text{and}\quad}{P}{\left({B}∣{A}\right)}={P}{\left({B}\right)}$$. Having soup and salad for lunch are not independent because $$\displaystyle{P}{\left({A}∣{B}\right)}={P}{\left({A}\right)}{\quad\text{and}\quad}{P}{\left({B}∣{A}\right)}={P}{\left({B}\right)}$$. Having soup and salad are not independent events because $$\displaystyle{P}{\left({A}∣{B}\right)}≠{P}{\left({A}\right)}{\quad\text{and}\quad}{P}{\left({B}∣{A}\right)}≠{P}{\left({B}\right)}$$. Having soup and salad for lunch are independent events because $$\displaystyle{P}{\left({A}∣{B}\right)}≠{P}{\left({A}\right)}{\quad\text{and}\quad}{P}{\left({B}∣{A}\right)}≠{P}{\left({B}\right)}$$

Upper level probability

### When a British travelling company reduces fares of a particular trip from London to Nottingham. A small taxi can carry four passengers. The time between calls for tickets is exponentially distributed with a mean of 30 minutes. Assume that each call orders one ticket. What is the probability that the taxi is filled in less than 3 hours from the time of the fare reduction?

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: $$\begin{array}{|c|c|} \hline Varieties & Location 1& Location 2& Location 3& Location 4& Location 5& Location 6 \\ \hline A& 35.3& 31.0& 32.7& 36.8& 37.2& 33.1 \\ \hline B& 30.7& 32.2& 31.4& 31.7& 35.0& 32.7\\ \hline C& 38.2& 33.4& 33.6& 37.1& 37.3& 38.2\\ \hline D& 34.9& 36.1& 35.2& 38.3& 40.2& 36.0\\ \hline E& 32.4& 28.9& 29.2& 30.7& 33.9& 32.1\\ \hline \end{array}$$ 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 $$\alpha\alpha=.05$$. b. How do the analysis of variance F test compare with the test in part a? Explain.

Upper level probability

### From a standard 52-card deck, what is the probability that a drawn card is a club or anything besides a face card?

Upper level probability