STAT243 LS: Intro to Probability and Statistics Final Exam, Mar 20, 2017 KEY

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1 STAT243 LS: Intro to Probability and Statistics Final Exam, Mar 20, 2017 KEY This is a 110-min exam. Students may use a page of note (front and back), and a calculator, but nothing else is allowed. 1. A normal distribution is: A) skewed right B) skewed left C) bi-modal D) symmetric 2. Scientists discovered a new group of proteins in an animal species. They found that the distribution of the number of amino acids these proteins were made of was approximately normal with mean 530 and standard deviation 80. About what percent of these new proteins will be between 370 and 690 amino acids long? A) 68% B) 95% C) 99.7% D) 32% 3. In a binomial setting, which of the following must NOT be true? A) There must be a fixed number of observations. B) The probability of success is the same for each observation. C) The observations must be dependent on each other. D) There are only two possible outcomes from each observation.

2 4. A lake sample was analyzed under a microscope to determine the number of clumps of alga cells per microscope field. It was found that on average there were three clumps. What is the expected number of clumps in four microscope fields? A) 3 B) 6 C) 12 D) A couple decides that they will keep having children until they have a boy. Successive births are independent of each other and the probability of having a boy is approximately Does this follow a binomial setting? A) Yes, each child has a 50% chance of being a boy. B) Yes, each child is independent of each other. C) No, the couple did not fix the number of observations In a matched pairs experiment, subjects pushed a button as quickly as they could after taking a caffeine pill and also after taking a placebo pill. The mean pushes per minute were 283 for the placebo and 311 for caffeine. 6. The numbers 283 and 311 are: A) statistics B) parameters C) confounded D) random numbers

3 7. The parameter values in this example are: A) 283 and 311 B) values less than 283 and greater than 311 C) two values between 283 and 311 D) unknown 8. The distribution of the averages calculated for repeated samples of size 10 of two-year-old male children is called A) a sampling distribution B) a population distribution C) a statistical distribution D) a parametric distribution 9. The gypsy moth is a serious threat to oak trees. Traps are placed throughout the state to detect moths. When checked, the traps contain varying numbers of moths, averaging 0.5 moths per trap. The distribution for the individual counts is strongly skewed with a standard deviation of 0.7. How would you describe the distribution for average number of moths per trap for 49 traps? A) symmetrical with mean = 0.5 and std dev = 0.7 B) symmetrical with mean = 0.5 and std dev = 0.1 C) skewed with mean = 0.5 and std dev = 0.1 D) skewed with mean = 0.5 and std dev = 0.7

4 10. If the circumference for women s heads is normally distributed with a mean of 22.2 and standard deviation of 1.4 inches, what proportion of women cannot use the ready-made helmets that exist for men due to the fact that circumferences of their heads are too small? Ready-made helmets range from to A) B) C) D) = 1.4 P(Z < 1.4) = The population of men with localized prostate tumor and a certain pretreatment has been studied. It was found that the probability of surviving at least 5 years is equal to 0.8. Consider a sample of 10 men with a localized prostate tumor and a pretreatment. Let X denote the number of these patients who will survive at least 5 years. 11. The probability that at least nine patients will survive at least 5 years is: A) B) C) D) P(X 9) = P(X = 9) + P(X = 10) = ( 10 9 ) (.8)9 (.2) 1 + ( ) (.8)10 (.2) 0 = =

5 12. The probability that all will survive at least 5 years is: A) B) C) D) P(X = 10) = ( ) (.8)10 (.2) 0 = The variance of the number of patients that will survive at least 5 years in a samples of n = 10 is: A) B) C) D) *.8*.2= As part of a promotion for a new type of cracker, free trial samples are offered to shoppers in a local supermarket. The probability that a shopper will buy a packet of crackers after tasting the free sample is Different shoppers can be regarded as independent trials. If X is the number among the next 100 shoppers who buy a packet of crackers after tasting a free sample, what is the approximate probability that at least 20 of the next 100 shoppers who sample the crackers will buy a pack? Hint: Compute the mean and the standard deviation, and then apply continuity-corrected normal approximation. A) 15% B) 25% C) 75% D) 55% P(20 X) = P(19.5 X) = P ( 100(. 2)(.8) X 20 ) = P( Z) 100(. 2)(.8) P( 0.13 Z) = P(Z < 0.13) =

6 A hospital delivers an average of 268 children per month. In the United States, one in every 500 babies is born with one or more extra fingers or toes. Let X be the count of babies born with one or more extra fingers or toes in a month at that hospital. 15. What is the variance of the number of babies born at that hospital in a month with an extra finger or toe? A) B) C) D) Note that Poisson random variable s mean and variance are the same. μ = 268 ( 1 ) = = σ What is the probability that at least two babies will be born with an extra finger or toe at that hospital in a month? Hint: P(X 2)? Use the formula: P(X = k) = e μ μ k You need to figure out μ and k in this formula. A) B) C) D) k! P(X 2) = 1 [P(X = 0) + P(X = 1)] = 1 [ e ! = 1 ( ) = = e ] 1!

7 17. Shelia s measured glucose level one hour after ingesting a sugary drink varies according to the Normal distribution with µ = 125 mg/dl and σ = 10 mg/dl. We took 16 separate measurements from Shelia. What is the blood glucose level L such that the probability is only that the average of 16 measurements is larger than L? A) B) C) D) σ X = = 2.5 L is the 97.5 th percentile of the sampling distribution of the sample mean. Since we know the mean of the sample mean is the population mean, 125, and the standard deviation of the sample mean is 5, L must be *2.5 =129.9, where 1.96 is the 97.5 th percentile of the standard normal distribution. 18. An insurance company knows that, in the entire population of millions of insured workers, the mean annual cost of workers compensation claims is µ= $439 per insured worker, and the standard deviation is σ = $20,000. The distribution of losses is strongly right-skewed: Most policies have no loss, but a few have large losses, up to millions of dollars. If the company sells 100 policies, what is the probability that the mean claim loss would be no greater than $500? A) B) C) D) σ X = ( ) = 2000; z = = P(Z.0305) P(Z.03) =

8 % of the American population has blood type O What is the probability that more than 37 out of the next 100 randomly sampled Americans will have blood type O+? A) 0.85 B) 0.50 C) 0.35 D) 0.25 Standard deviation of the sample proportion is equal to /100 = Sample proportion,.37 = 37/100, is z = ( )/ = 0 standard deviation up from the mean proportion,.37, which implies P(Z>0) = 0.5 due to symmetry. 20. What is the probability that more than 370 out of the next 1000 randomly sampled Americans will have blood type O+? A) B) C) D) Standard deviation of the sample proportion is equal to /1000 = Sample proportion,.37 = 370/1000, is ( )/ = 0 standard deviation up from the mean proportion,.37. From the normal probability table, P(Z>0) = 0.5 due to symmetry.