Download M155ST4.5Ch56a.notebook 1 March 15, 2010

M155ST4.5­Ch5­6a.notebook
March 15, 2010
MAT 155
Sample Problems for Test 2
Sections 4.5­4.7 Probability
Chapter 5 Discrete Probability Distributions
Chapter 6 Normal Probability Distribution
2.
Calculate the following and SHOW work in formula.
(a) 7P3 =
(d) 4! =
Mar 5­7:36 AM
3. Of 9 people, (a) you select a committee of 3, and (b) you elect 3 officers (president, vice­president, and secretary/treasurer). SHOW your work for each calculation.
(b) 5C2 =
(d) =
Mar 5­8:22 AM
4. Does P(x) = x/5 (where x can take on the values of 0, 1, 2, 3) describe a probability distribution? Fill in the table below using fractions for P(x). Explain your answer.
(a) How many different 3­person committees are possible?
(b) How many different slates of candidates for officers are possible?
Mar 5­8:23 AM
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8. Use the Binomial Probability Formula to find the probability of 4 successes in 7 trials when the probability of success is 0.2. Show your work and do not round off your answer.
5. To settle a paternity suit, two different people are given blood tests. If x is the number having group A blood, then x can be 0, 1, or 2, and the corresponding probabilities are 0.35, 0.48, and 0.17, respectively (based on data from the Greater New York Blood Program). Fill in the table below. Determine whether the above information describes a probability distribution. If it does not, explain why. If it does, find its mean.
9. Use the calculator function “binomcdf(” to find the probability of at most 4 successes in 7 trials when the probability of success is 0.2. Write down what you entered into your calculator and the answer. Do not round off your answer.
Mar 5­8:24 AM
Mar 5­8:26 AM
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M155ST4.5­Ch5­6a.notebook
March 15, 2010
10. Use StatDisk to answer the following questions for a binomial probability distribution when n = 5 and p = 0.4. Round off your answers to 5­decimal places.
(a) P(x = 3) =
(b) P(x ≤ 4) = (c) P(x ≥ 2) = (d) P(x < 2) = 2. Suppose the MicroSort method of gender­selection is used with 100 couples, each of whom will have 1 baby. Assume that the result is 68 girls among the 100 babies.
(a) Assuming that the MicroSort gender­selection method has no effect, find the mean and standard deviation for the numbers of girls in groups of 100 randomly selected babies.
(b) Interpret the values from part (a) to determine whether this result of 68 girls among 100 babies supports a claim that the MicroSort method of gender­selection is effective.
Mar 5­8:27 AM
3. Right­Handed People. Ninety percent of American adults are right­handed. A statistics class has 20 students in attendance. (HINT: This satisfies binomial conditions.)
(a) Find the mean and standard deviation for the number of right­handed students in such classes of 20 students.
Mar 5­8:28 AM
4. For the binomial probability distribution with n = 1984 and p = 0.75, find (a) the mean, (b) the standard deviation, (c) the minimum usual value µ ­ 2σ, and ﴾d﴿ the maximum usual value µ + 2σ. SHOW your work and round off to two­decimal places.
(b) Would it be unusual to survey a class of 20 students and find that 15 of them are right­handed? Why or why not?
Mar 5­8:29 AM
5. Assume that the Poisson distribution applies with µ = 10. Find P(x=4). Insert the numbers into the Poisson distribution formula and find the answer for P(x=4). [SHOW your work. You may use a calculator or computer to do the computations.]
Mar 5­8:31 AM
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2. Assume that z scores are normally distributed with a mean of 0 and a standard deviation of 1. Sketch and properly label a bell­shaped curve for parts (a) and (b). Show proper work to justify your answers.
(a) If P(0 < z < a) = 0.3907, find a.
(b) If P(x > b) = 0.9922, find b.
Mar 5­8:32 AM
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M155ST4.5­Ch5­6a.notebook
3. Based on survey data from Gordon, Churchill, et al., women have forward grip reaches that are normally distributed with a mean of 27.0 inches and a standard deviation of 1.3 inches. Design engineers decided that the CD player should be placed so that it is within the forward grip reach of 95% of women. Find the forward grip reach that separates the 95% of women with the longest reach from the others. Use Table A­2. Make a sketch, label it properly, and show your work to justify your answer.
March 15, 2010
5. Assume that men’s weights are normally distributed with µ = 172 lbs and σ = 29 lbs. Draw and label an appropriate sketch for each problem. Use calculator, Table A­2, or computer. SHOW your work, function used and values entered for each problem.
(a) If a man is randomly selected, find the probability that he has a weight between 160 lbs and 180 lbs.
(b) If 4 men are randomly selected, find the probability that they have a mean weight between 160 lbs and 180 lbs.
Mar 5­8:33 AM
1. The probability of flu symptoms for a person not receiving any treatment is 0.019. In a clinical study, 863 patients were given a treatment of 10mg of medicine in tablet form, and 19 of those patients experienced flu symptoms. Assuming that these tablets have no effect on flu symptoms, estimate the probability that at least 23 of the 863 people experience flu symptoms. Use binomcdf and use normalcdf with continuity correction factor.
Mar 5­8:34 AM
Mar 5­8:33 AM
2. Scores on a test are normally distributed with a mean of 82.9 and a standard deviation of 4.9. Find the value of P85 to three­significant digits.
D) 90.8 C) 88.0 B) 0.880
A) 0.908 Mar 5­8:35 AM
4. Estimate the indicated probability by using the normal distribution as an approximation to the binomial distribution. In one county, the conviction rate for speeding is 75%. Estimate the probability that of the next 160 speeding summonses issued, there will be at most 130 convictions. A) 0.0276 B) 0.9661 C) 0.9724 D) 0.0339 Mar 15­9:34 AM
Mar 5­8:36 AM
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M155ST4.5­Ch5­6a.notebook
14. When you give a casino $10 for a bet on the “pass line” in the game of craps, there is a 244/495 probability that you will win $10 and a 251/495 probability that you will lose $10. What is your expected value?
March 15, 2010
15. Given that n = 6 and p = 0.95 (in a binomial probability distribution), find the following probabilities:
The probability of exactly 2 successes.
a.
The probability of at least 2 successes.
b.
Mar 5­7:56 AM
16. The rates of on­time flights for commercial jets are continuously tracked by the U.S. Department of Transportation. Recently, Southwest Air had the best rate with 80% of its flights arriving on time. A test is conducted by randomly selecting 15 Southwest flights and observing whether they arrive on time.
Find the probability that exactly 8 flights arrive on time.
a.
Would it be unusual for Southwest to have 6 flights arrive late? Why or why not?
b.
Mar 5­7:55 AM
17. Several students are unprepared for a multiple­choice test with 20 questions, and all of their answers are guesses. Each question has four possible answers, and only one of them is correct.
Find the mean for the number of correct answers for such students.
a.
Find the standard deviation for the number of correct answers for such students.
b.
Would it be unusual for a student to pass by guessing and getting at least 12 correct c.
answers? Why or why not?
Mar 5­7:52 AM
18. Mars, Inc., claims that 10% of its M&M plain candies are blue, and a sample of 150 such candies is randomly selected. Find the mean for the number of blue candies in such groups of 150.
a.
Find the standard deviation for the number of blue candies in such groups of 150.
b.
Mar 5­7:53 AM
Mar 5­7:53 AM
For Problems 1 and 2, assume that voltages in a circuit vary between 5 volts and 11 volts, and voltages are spread evenly over the range of possibilities (Uniform Distribution). Find the probability of the given range of voltage levels. [Make a sketch for each problem.]
1.
Greater than 9 volts
2.
Between 5.5 volts and 10 volts Mar 5­7:59 AM
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M155ST4.5­Ch5­6a.notebook
March 15, 2010
For Problems 3, 4, 5, and 6, assume that the readings on the thermometers are normally distributed with a mean of 0°C and a standard deviation of 1.00°C. [Make a sketch for each problem.]
For Problems 3, 4, 5, and 6, assume that the readings on the thermometers are normally distributed with a mean of 0°C and a standard deviation of 1.00°C. [Make a sketch for each problem.]
3. Find the indicated probability where z is the reading in degrees: P(z < 1.645)
5. Find the temperature reading corresponding to P20, the 20th percentile.
4. Find the indicated probability where z is the reading in degrees: P(1.96 < z < 2.33)
6. If 12% of the thermometers are rejected because they have readings that are too high, but all other thermometers are acceptable, find the reading that separates the rejected thermometers from the others. [Make a sketch for each problem.]
Mar 5­8:08 AM
For Problems 7, 8, 9, and 10, assume that women’s weights are normally distributed with a mean given by µ = 143 lbs and standard deviation given by σ = 29 lbs. Also, assume that a woman is randomly selected. Let x = weight in pounds.
Mar 5­8:09 AM
For Problems 7, 8, 9, and 10, assume that women’s weights are normally distributed with a mean given by µ = 143 lbs and standard deviation given by σ = 29 lbs. Also, assume that a woman is randomly selected. Let x = weight in pounds.
9. Find the sixth decile, D6, which is the weight separating the bottom 60% from the top 40%.
7. Find the indicated probability: P(150 < x < 180)
10. a) If 1 woman is randomly selected, find the probability that her weight is above 140 lb.
b) If 100 women are randomly selected, find the probability that they have a mean weight greater than 140 lb.
8. Find the indicated probability: P(x < 186.5)
Mar 5­8:11 AM
11. The U.S. Army requires women’s heights to be between 58 in. and 80 in. Find the percentage of women meeting that height requirement. Are many women being denied the opportunity to join the Army because they are too short or too tall? [Assume that heights of women are normally distributed with a mean of µ = 63.6 in. and a standard deviation of σ = 2.5 in.]
Mar 5­8:16 AM
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12. The lengths of pregnancies are normally distributed with a mean of 268 days and a standard deviation of 15 days. If we stipulate that a baby is premature if born at least three weeks early, what percentage of babies is born prematurely?
Mar 5­8:17 AM
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M155ST4.5­Ch5­6a.notebook
For Problems 13 and 14, use the fact that IQ scores are normally distributed with a mean of 100 and a standard deviation of 15. [Make a sketch for each problem.]
13. a) People are considered to be “intellectually very superior” if their score is above 130. What percentage of people fall into that category?
b) If we redefine the category of “intellectually very superior” to be scores in the top 2%, what does the minimum score become?
Mar 5­8:18 AM
March 15, 2010
For Problems 13 and 14, use the fact that IQ scores are normally distributed with a mean of 100 and a standard deviation of 15. [Make a sketch for each problem.]
14. If 25 people are randomly selected for an IQ test, find the probability that their mean IQ score is between 95 and 105.
Mar 5­8:19 AM
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