Suppose X has a hypergeometric distribution with N 100, n 4, and K 20. Determine the following: (a) (b) (c) (d) Determine the mean and variance of X.
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2-1
SAMPLE SPACES AND EVENTS
2-2
INTERPRETATIONS OF PROBABILITY
2-3
ADDITION RULES
2-4
CONDITIONAL PROBABILITY
2-5
MULTIPLICATION AND TOTAL PROBABILITY RULES
2-6
INDEPENDENCE
2-7
BAYES THEOREM
2-8
RANDOM VARIABLES
3-1
DISCRETE RANDOM VARIABLES
3-2
PROBABILITY DISTRIBUTIONS AND PROBABILITY MASS FUNCTIONS
3-3
CUMULATIVE DISTRIBUTION FUNCTIONS
3-4
MEAN AND VARIANCE OF A DISCRETE RANDOM VARIABLE
3-5
DISCRETE UNIFORM DISTRIBUTION
3-6
BINOMIAL DISTRIBUTION
3-7
GEOMETRIC AND NEGATIVE BINOMIAL DISTRIBUTIONS
3-8
HYPERGEOMETRIC DISTRIBUTION
3-9
POISSON DISTRIBUTION
4-10
ERLANG AND GAMMA DISTRIBUTIONS
4-11
WEIBULL DISTRIBUTION
4-12
LOGNORMAL DISTRIBUTION
4-2
PROBABILITY DISTRIBUTIONS AND PROBABILITY DENSITY FUNCTIONS
4-3
CUMULATIVE DISTRIBUTION FUNCTIONS
4-4
MEAN AND VARIANCE OF A CONTINUOUS RANDOM VARIABLE
4-5
CONTINUOUS UNIFORM DISTRIBUTION
4-6
NORMAL DISTRIBUTION
4-7
NORMAL APPROXIMATION TO THE BINOMIAL AND POISSON DISTRIBUTIONS
4-8
CONTINUITY CORRECTIONS TO IMPROVE THE APPROXIMATION
4-9
EXPONENTIAL DISTRIBUTION
5-1
TWO DISCRETE RANDOM VARIABLES
5-10
CHEBYSHEVS INEQUALITY (CD ONLY)
5-2
MULTIPLE DISCRETE RANDOM VARIABLES
5-3
TWO CONTINUOUS RANDOM VARIABLES
5-4
MULTIPLE CONTINUOUS RANDOM VARIABLES
5-5
COVARIANCE AND CORRELATION
5-6
BIVARIATE NORMAL DISTRIBUTION
5-7
LINEAR COMBINATIONS OF RANDOM VARIABLES
5-8
FUNCTIONS OF RANDOM VARIABLES (CD ONLY)
5-9
MOMENT GENERATING FUNCTIONS (CD ONLY)
6-1
DATA SUMMARY AND DISPLAY
6-3
STEM-AND-LEAF DIAGRAMS
6-4
FREQUENCY DISTRIBUTIONS AND HISTOGRAMS
6-5
BOX PLOTS
6-6
TIME SEQUENCE PLOTS
6-7
PROBABILITY PLOTS
6-8
MORE ABOUT PROBABILITY PLOTTING (CD ONLY)
7-2
GENERAL CONCEPTS OF POINT ESTIMATION
7-3
METHODS OF POINT ESTIMATION
7-5
SAMPLING DISTRIBUTIONS OF MEANS
8-2
CONFIDENCE INTERVAL ON THE MEAN OF A NORMAL DISTRIBUTION, VARIANCE KNOWN
8-3
CONFIDENCE INTERVAL ON THE MEAN OF A NORMAL DISTRIBUTION, VARIANCE UNKNOWN
8-4
CONFIDENCE INTERVAL ON THE VARIANCE AND STANDARD DEVIATION OF A NORMAL POPULATION
8-5
A LARGE-SAMPLE CONFIDENCE INTERVAL FOR A POPULATION PROPORTION
8-6
A PREDICTION INTERVAL FOR A FUTURE OBSERVATION
8-7
TOLERANCE INTERVALS FOR A NORMAL DISTRIBUTION
9-1
HYPOTHESIS TESTING
9-2
TESTS ON THE MEAN OF A NORMAL DISTRIBUTION, VARIANCE KNOWN
9-3
TESTS ON THE MEAN OF A NORMAL DISTRIBUTION, VARIANCE UNKNOWN
9-4
HYPOTHESIS TESTS ON THE VARIANCE AND STANDARD DEVIATION OF A NORMAL POPULATION
9-5
TESTS ON A POPULATION PROPORTION
9-7
TESTING FOR GOODNESS OF FIT
10-3
INFERENCE FOR THE DIFFERENCE IN MEANS OF TWO NORMAL DISTRIBUTIONS, VARIANCES UNKNOWN
10-4
INFERENCE FOR THE DIFFERENCE IN MEANS OF TWO NORMAL DISTRIBUTIONS, VARIANCES UNKNOWN
10-5
INFERENCES ON THE VARIANCES OF TWO NORMAL POPULATIONS
10-6
INFERENCE ON TWO POPULATION PROPORTIONS
10-7
SUMMARY TABLE FOR INFERENCE PROCEDURES FOR TWO SAMPLES
10.2
INFERENCE FOR A DIFFERENCE IN MEANS OF TWO NORMAL DISTRIBUTIONS, VARIANCES KNOWN
11-11
CORRELATION
11-2
SIMPLE LINEAR REGRESSION
11-5
HYPOTHESIS TESTS IN SIMPLE LINEAR REGRESSION
11-7
PREDICTION OF NEW OBSERVATIONS
11-8
ADEQUACY OF THE REGRESSION MODEL
12-1
MULTIPLE LINEAR REGRESSION MODEL
12-2
MULTIPLE LINEAR REGRESSION MODEL
12-3
CONFIDENCE INTERVALS IN MULTIPLE LINEAR REGRESSION
12-5
MODEL ADEQUACY CHECKING
12-6
ASPECTS OF MULTIPLE REGRESSION MODELING
13-2
THE COMPLETELY RANDOMIZED SINGLE-FACTOR EXPERIMENT
13-4
RANDOMIZED COMPLETE BLOCK DESIGN
14-4
TWO-FACTOR FACTORIAL EXPERIMENTS
14-5
GENERAL FACTORIAL EXPERIMENTS
14-7
2k FACTORIAL DESIGNS
14-8
BLOCKING AND CONFOUNDING IN THE 2k DESIGN
14-9
FRACTIONAL REPLICATION OF THE 2k DESIGN
15-2
SIGN TEST
15-3
WILCOXON SIGNED-RANK TEST
15-4
WILCOXON RANK-SUM TEST
15-5
NONPARAMETRIC METHODS IN THE ANALYSIS OF VARIANCE
16-10
CUMULATIVE SUM CONTROL CHART
16-12
IMPLEMENTING SPC
16-5
x AND R OR S CONTROL CHARTS
16-6
CONTROL CHARTS FOR INDIVIDUAL MEASUREMENTS
16-7
PROCESS CAPABILITY
16-8
ATTRIBUTE CONTROL CHARTS
16-9
CONTROL CHART PERFORMANCE
33-3
THE RANDOM-EFFECTS MODEL
Textbook Solutions for Applied Statistics and Probability for Engineers
Chapter 3-8 Problem 3-91
Question
3-91. A company employs 800 men under the age of 55. Suppose that 30% carry a marker on the male chromosome that indicates an increased risk for high blood pressure. (a) If 10 men in the company are tested for the marker in this chromosome, what is the probability that exactly 1 man has the marker? (b) If 10 men in the company are tested for the marker in this chromosome, what is the probability that more than 1 has the marker?
Solution
The first step in solving 3-8 problem number 6 trying to solve the problem we have to refer to the textbook question: 3-91. A company employs 800 men under the age of 55. Suppose that 30% carry a marker on the male chromosome that indicates an increased risk for high blood pressure. (a) If 10 men in the company are tested for the marker in this chromosome, what is the probability that exactly 1 man has the marker? (b) If 10 men in the company are tested for the marker in this chromosome, what is the probability that more than 1 has the marker?
From the textbook chapter HYPERGEOMETRIC DISTRIBUTION you will find a few key concepts needed to solve this.
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full solution
full solution
Title
Applied Statistics and Probability for Engineers 3
Author
Douglas C. Montgomery, George C. Runger
ISBN
9780471204541
3-91. A company employs 800 men under the age of 55. Suppose that 30% carry a marker on
Chapter 3-8 textbook questions
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Chapter 3: Problem 3 Applied Statistics and Probability for Engineers 3
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Chapter 3: Problem 3 Applied Statistics and Probability for Engineers 3
3-87. Suppose X has a hypergeometric distribution with N 20, n 4, and K 4. Determine the following:
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Chapter 3: Problem 3 Applied Statistics and Probability for Engineers 3
Suppose X has a hypergeometric distribution with N 10, n 3, and K 4. Sketch the probability mass function of X.
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Chapter 3: Problem 3 Applied Statistics and Probability for Engineers 3
3-89. Determine the cumulative distribution function for X in Exercise 3-88.
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Chapter 3: Problem 3 Applied Statistics and Probability for Engineers 3
A lot of 75 washers contains 5 in which the variability in thickness around the circumference of the washer is unacceptable. A sample of 10 washers is selected at random, without replacement. (a) What is the probability that none of the unacceptable washers is in the sample? (b) What is the probability that at least one unacceptable washer is in the sample? (c) What is the probability that exactly one unacceptable washer is in the sample?
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Chapter 3: Problem 3 Applied Statistics and Probability for Engineers 3
3-91. A company employs 800 men under the age of 55. Suppose that 30% carry a marker on the male chromosome that indicates an increased risk for high blood pressure. (a) If 10 men in the company are tested for the marker in this chromosome, what is the probability that exactly 1 man has the marker? (b) If 10 men in the company are tested for the marker in this chromosome, what is the probability that more than 1 has the marker?
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Chapter 3: Problem 3 Applied Statistics and Probability for Engineers 3
Printed circuit cards are placed in a functional test after being populated with semiconductor chips. A lot contains 140 cards, and 20 are selected without replacement for functional testing. (a) If 20 cards are defective, what is the probability that at least 1 defective card is in the sample? (b) If 5 cards are defective, what is the probability that at least 1 defective card appears in the sample?
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Chapter 3: Problem 3 Applied Statistics and Probability for Engineers 3
3-93. Magnetic tape is slit into half-inch widths that are wound into cartridges. A slitter assembly contains 48 blades. Five blades are selected at random and evaluated each day for sharpness. If any dull blade is found, the assembly is replaced with a newly sharpened set of blades. (a) If 10 of the blades in an assembly are dull, what is the probability that the assembly is replaced the first day it is evaluated? (b) If 10 of the blades in an assembly are dull, what is the probability that the assembly is not replaced until the third day of evaluation? [Hint: Assume the daily decisions are independent, and use the geometric distribution.] (c) Suppose on the first day of evaluation, two of the blades are dull, on the second day of evaluation six are dull, and on the third day of evaluation, ten are dull. What is the probability that the assembly is not replaced until the third day of evaluation? [Hint: Assume the daily decisions are independent. However, the probability of replacement changes every day.]
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Chapter 3: Problem 3 Applied Statistics and Probability for Engineers 3
A state runs a lottery in which 6 numbers are randomly selected from 40, without replacement. A player chooses 6 numbers before the states sample is selected. (a) What is the probability that the 6 numbers chosen by a player match all 6 numbers in the states sample? (b) What is the probability that 5 of the 6 numbers chosen by a player appear in the states sample? (c) What is the probability that 4 of the 6 numbers chosen by a player appear in the states sample? (d) If a player enters one lottery each week, what is the expected number of weeks until a player matches all 6 numbers in the states sample?
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Chapter 3: Problem 3 Applied Statistics and Probability for Engineers 3
Continuation of Exercises 3-86 and 3-87. (a) Calculate the finite population corrections for Exercises 3-86 and 3-87. For which exercise should the binomial approximation to the distribution of X be better? (b) For Exercise 3-86, calculate and assuming that X has a binomial distribution and compare these results to results derived from the hypergeometric distribution. (c) For Exercise 3-87, calculate and assuming that X has a binomial distribution and compare these results to the results derived from the hypergeometric distribution.
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Chapter 3: Problem 3 Applied Statistics and Probability for Engineers 3
Use the binomial approximation to the hypergeometric distribution to approximate the probabilities in Exercise 3-92. What is the finite population correction in this exercise?
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