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This is the Exam of Probability which includes Maximum, Hazard Rate Function, Continuous Random Variable, Density Function, Definition, Compute, Geometric, Geometric Random Variable etc. Key important points are: Marking, Time Carefully, Each Statement, True, Box Plot, Skewed, Unbiased Estimator, Formula, Sample Variance, Observations
Typology: Exams
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Part I Instructions: Respond to all of the questions below. For full credit, you must justify your answers. Calculators are not permitted.
Points Possible Points Earned
1 5 2 3 3 3
4 2 5 2
Note: This exam consists of 2 parts. Part II contains 9 questions, is worth 85 points, and calculators are permitted. When you complete Part I, please turn it in and pick up Part II. Be sure to manage your time carefully during this exam.
1
(1) Indicate whether each statement is true (by marking a “T”) or false (by marking an “F”). You do not need to explain your answer. P (A ∪ B) = P (A) + P (B)
P (A|B) = P^ (B P|A (B)P)^ (A) n r
= n^
P (^) r r! The data for the following box plot is skewed to the left:
A statistic θˆ is an unbiased estimator if: E(θˆ) = θ.
(2) Short Answer: For these questions, you do not need to justify your answer (al- though you may do so if you wish). (a) Complete the formula for the sample variance (hint: your respond should de- pend on a set of n observations: x 1 , x 2 ,... , x (^) n ):
s 2 =
(b) In this course, we learned four ways of looking at a set of data to search for evidence of nonnormality. State them. (i)
(ii)
(iii)
(iv)
(c) Let the random variable Z have a standard normal distribution. Find the following probability:
P (Z < − 0 .76) =
(4) During the spring, many rivers are monitored very carefully in an attempt to warn residents of an impending flood. The depth (in feet) of the Susquehanna River at the Bloomsburg bridge is measured and reported daily. In a random sample of depths, x = 16. 7 , s = 2.1, and the shape of the distribution is approximately normal. (a) Approximately what proportion of depths is between 14.6 and 18.8?
(b) Approximately what proportion of depths is less than 14.6?
(5) In your own words, explain the difference between a parameter and a statistic. Be sure to use complete sentences.
Name:
Part II Instructions: Respond to all of the questions below. For full credit, you must justify your answers. For questions where you use a calculator, be sure to show work by writing down what you typed into your calculator.
Points Possible Points Earned 6 9
7 12 8 7 9 4
10 11 11 10 12 9
13 14 14 9
(7) The McPherson Middle School in Clyde, Ohio, is examining its school lunch pro- gram. A survey asked 2200 students about their lunch types and how they got to school in the morning. The following (partial) two-way table is assumed to represent the entire student body.
(a) Complete the two-way table. (b) Suppose a student is randomly selected. What is the probability that the stu- dent carries a lunch and gets to school by car?
(c) Suppose the student takes the bus to school. What is the probability that the student buys lunch?
(d) Suppose the student does not walk to school. What is the probability that the student carries a lunch?
(8) The best-selling basketball sneaker of all time is the Converse 8220 Chuck Taylor 8221 All Star. Suppose Chuck Taylor 8221s are sold in Holabird Sports and are available in sizes from 7 to 13. The size purchased is a random variable with probability distribution given in the table below. x 7 8 9 10 11 12 13 p(x) 0.05 0.08 0.10 0.14 0.28 0.21 0. (a) Find the mean and variance of the sneaker size purchased.
(b) Suppose three randomly selected customers each purchase a pair of these sneak- ers. What is the probability that exactly two of the three buy size 11 sneakers?
(11) Tropical rainforests cover approximately 6% of the Earth’s surface. The temperature in a typical rainforest ranges from 68 to 93◦^ F, and the humidity is usually between 77% and 88%. The mean amount of rain per year in any one rainforest is μ = 155 inches with standard deviation σ = 35 inches. Suppose 30 rainforests are selected at random and the amount of rain per year is recorded for each. (a) Find the distribution of the mean rainfall per year for the 30 rainforests.
(b) What is the probability that the mean rainfall is more than 170 inches?
(c) Find a value r such that the probability that the mean rainfall is less than r is 0.001.
(12) The latest survey by the National Golf Foundation suggests that 7.4% of all adult golfers play left-handed. In a random sample of 1250 golfers from around the coun- try, 101 indicated that they play left-handed. Find a 95% confidence interval for the population proportion of golfers who play left-handed (and, as always, be sure to show all work).
(c) Find the probability of a type II error if the true population mean moisture content is 12.2%. That is, find β(12.2).
(14) A piston in a particular 12-cylinder diesel engine is manufactured to have diameter 13 mm. Any larger or smaller diameter will cause immediate costly damage to the engine. Every half hour, 10 finished pistons are selected and the diameter of each (in mm) is carefully measured. If there is any evidence that the mean diameter is different from 13 mm, the manufacturing process is stopped and the machinery is inspected. The quality-control inspector uses a significance level of α = 0. 05 , and supposes that the underlying population of pistons is normal. (a) Suppose the sample mean is x = 12.89 and the sample standard deviation is s = 0. 96. Should the manufacturing process be stopped?
(b) If you were buying these pistons, would you like the manufacturer to use a smaller or larger significance level? Justify your answer (and please use complete sentences).
