Exam Practice # 2 (Coverage Ch4-Ch5) NOTE: These are just Practice Problems for Exam # 2. This is NOT meant to look just like the test, and it is NOT the only thing that you should study. Make sure you know all the material from the lecture notes, quizzes, suggested homework problems and the corresponding chapters in the book. Use the following to answer questions 1-2: Many residents of suburban neighborhoods own more than one car but consider one of their cars to be the main family vehicle. The age of these family vehicles can be modeled by a normal distribution with mean 2 years and standard deviation 6 months. 1. What percentage of family vehicles is between 1 and 3 years old?
2.
What is the standardized value for a family vehicle that is 3 years and 3 months old?
3.
The time to complete a standardized exam is approximately normal with a mean of 70 minutes and a standard deviation of 10 minutes. Using the 68-95-99.7 rule, what percentage of students will complete the exam in under an hour?
1|P age
Use the following to answer questions 4-5: The temperature at any random location in a kiln used for manufacturing bricks is normally distributed with a mean of 1000F and a standard deviation of 50F. 4. If bricks are fired at a temperature above 1125F, they will crack and must be discarded. If the bricks are placed randomly throughout the kiln, what is the percentage of bricks that crack during the firing process?
5.
When glazed bricks are put in the oven, if the temperature is below 900F they will discolor. If the bricks are placed randomly throughout the kiln, what percentage of glazed bricks will discolor?
6.
The scores on a university examination are normally distributed with a mean of 62 and a standard deviation of 11. If the bottom 5% of students will fail the course, what is the lowest mark that a student can have and still be awarded a passing grade?
2|P age
7.
The preparation time to mail envelopes with a weekly report to all executives in a company has a normal distribution with a mean of 35 minutes and a standard deviation of 2 minutes. On 95% of such occasions the mailing preparation takes less than x minutes. What is the value of x?
8.
A soft-drink machine can be regulated so that it discharges an average of ounces per cup. If the ounces of fill are normally distributed with a standard deviation of 0.4 oz, what value should be set at so that 6-oz. cups will overflow only 2% of the time?
Use the following to answer questions 9-10: A company produces packets of soap powder labeled Giant Size 32 oz. The actual weight of soap powder in such a box has a normal distribution with a mean of 33 oz. and a standard deviation of 0.7 oz. To avoid dissatisfied customers, a box of soap is considered underweight if it weighs less than 32 oz. To avoid losing money, the top 5% (the heaviest 5%) is labeled overweight. 9. What proportion of boxes is underweight?
10.
How heavy does a box have to be in order for the box to be labeled overweight?
3|P age
Use the following to answer questions 11-13: Chocolate bars produced by a certain machine are labeled with 8.0 oz. The distribution of the actual weights of these chocolate bars is normal with a mean of 8.1 oz. and a standard deviation of 0.1 oz. A chocolate bar is considered underweight if it weighs less than 8.0 oz. 11. What proportion of chocolate bars weighs less than 8.0 oz?
12.
What proportion of chocolate bars weighs between 8.2 and 8.3 oz.?
13.
How should the chocolate bar wrappers be labeled so that only 1% of such bars are underweight?
4|P age
Use the following to answer questions 14-17: When figure skaters need to find a partner for pair figure skating, it is important to find a partner who is compatible in weight. The weight of figure skaters can be modeled by a normal distribution. For male skaters, the mean is 170 lbs. with a standard deviation of 10 lbs. For female skaters, the mean is 110 lbs. with a standard deviation of 5 lbs. Let the random variable X = the weight of female skaters and the random variable Y = the weight of male skaters. 14. What is P(X < 100)?
15.
Approximately 90% of the male skaters weigh more than how many pounds?
16.
The weight of a pair of figure skaters (a male and a female) can be thought of as a new random variable. Let the random variable W = X + Y. What is the mean of this new random variable W?
17.
Suppose we consider the weights of the male partner and the female partner to be independent. What is the standard deviation of the random variable W?
5|P age
Use the following to answer questions 18-20: The scores of individual students on the American College Testing (ACT) Program composite college entrance examination have a normal distribution with mean 18.6 and standard deviation 6.0. At North side High, 36 seniors take the test. Assume the scores at this school have the same distribution as national scores. 18. What is the mean of the sampling distribution of the sample mean score for a random sample of 36 students?
19.
What is the deviation standard of the sampling distribution of the sample mean score for a random sample of 36 students?
20.
What is the sampling distribution of the sample mean score for a random sample of 36 students?
6|P age
Use the following to answer questions 21-23: Chocolate bars produced by a certain machine are labeled 8.0 oz. The distribution of the actual weights of these chocolate bars is claimed to be normal with a mean of 8.1 oz. and a standard deviation of 0.1 oz. 21. A quality control manager initially plans to take a simple random sample of size n from the production line. If he were to double his sample size (to 2n), by what factor would the standard deviation of the sampling distribution of X change?
22.
The quality control manager plans to take a simple random sample of size n from the production line. How big should n be so that the sampling distribution of X has standard deviation 0.01 oz.?
23.
If the quality control manager takes a simple random sample of ten chocolate bars from the production line, what is the probability that the sample mean weight of the ten sampled chocolate bars will be less than 8.0 oz.?
24.
The lifetime of a particular circuit has an exponential distribution with mean 2 years. And we assume the circuit is now four years old and is still functioning. Find the probability that it functions for more than additional three more years.
7|P age
25. 26. 27.
If X is a continuous random variable, and c is any number, then P(X = c) = __________. If the assembly time for a product is uniformly distributed between 15 to 20 minutes, then probability of assembling product between 16 to 18 is __________. A continuous distribution is said to be __________ if the graph of its probability density function to the left of some point is a mirror image of the graph to the right of that point. If Z is a standard normal random variable with cumulative distribution function ( z ) , then (1.65) (1.65) = __________. The 90th percentile of the standard normal distribution is approximately z = __________. If the population distribution of a variable is approximately normal, then about __________% of the values are within one standard deviation of the mean. When X is a standard gamma random variable, the cumulative distribution function of X is called the __________ gamma function. The standard deviation of a random variable X having the exponential distribution with parameter .5 is __________. The variance of a random variable X having the chi-squared distribution with 5 degrees of freedom is 2 __________ . Two discrete random variables X and Y are said to be __________ if for every pair of x and y values, their __________ probability mass function (pmf) is the product of their __________ pmfs. The __________ gives a point estimate of the population standard deviation. A __________, such as sample mean X or sample standard deviation S, is any quantity whose value can be calculated from sample data. Let X1 , X 2 ,....., X 25 be a random sample from a population with mean of 60 and standard deviation of 16. Then x = __________. If the sample size n is __________ than 30, the central limit theorem can be used. If are independent random variables with variances 12 , 22 ,...., n2 , and the random variable Y a1 X1 a2 X 2 ..... an X n , then Var(Y) = __________.
28.
29. 30. 31. 32.
33.
34.
35. 36. 37.
38. 39.
8|P age
Use the following to answer questions 40-44:
( x 2 y) / 36, x y, x 1, 2, 3, y 1, 2, 3 Let X and Y have the joint pmf p(x,y) = . 0 , otherwise
40. Make the joint pmf table for (X,Y). p(x,y) 1 x 2 3 41. Find the marginal pmf of Y. 1 Y 2 3
42.
Find the conditional pmf of X given that Y=2, i.e., pX|Y(x|2).
43.
Find the mean of Y.
44.
Find the standard deviation of Y.
9|P age
Use the following to answer questions 45-49:
15x 2 y, 0 x y 1 Let X and Y have the joint pdf f(x,y) = . 0 , otherwise
45. Find Find the marginal pdf of X.
46.
Find Find the mean of X.
47.
Find Find E(XY).
48.
Find Find P(X+Y< ).
49.
Find Find P(YX< ).
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Use the following to answer questions 50-54: Let X~exp(4) and Y~exp(4). And X and Y are independent. Let T=X+Y. 50. Find Find the cdf of T, FT(t).
51.
Find Find the pdf of T, fT(t).
52.
Find Find P(X+Y> ), i.e., P(T> ).
53.
Find Find E(XY).
54.
Find Find the standard deviation of T.
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Answer Key
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 95% 2.5 16% .0062 .0228 44 (about) 38.29 5.18 .0766 34.15 .159 .136 7.87 .0228 157.2 280 11.18 18.6 1 Exactly normal. 1/sqrt(2) 100 .0008 .223 0 .4 symmetric .901 1.28 68 incomplete 2 10 independent, joint, marginal sample standard deviation statistic 3.2 larger
2 2 2 2 2 2 a1 1 a2 2 an n
In class In class In class 13/6 .799 In class 5/8 15/28 5/2048 9/128 In class In class .406 .0625 1/sqrt(8)=.354
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