Download Ch. 5 Topics

Ch. 5 Topics
You are only responsible for sections 5.1 and 5.2
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Sampling distribution
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Population distribution
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Count (X)
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Sample proportion (X/n = p-hat)
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The Binomial Setting
o Fixed number of observations (n)
o All observations are independent
o Each observation is either a success or failure
o Probability of success (p) is the same for all observations
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Binomial Distribution
o Distribution of the count X of successes in a binomial setting
o B(n, p)
o n and p are parameters
o n = number of observations taken/sample size
o p = X/n
o X can be any whole number from 0 to n
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Binomial Mean and St. Deviation
o μX = n*p
o σX = √(n*p*(1-p))
o Using calculator functions to find probabilities:
 binompdf(n, p, X) to find P(X = k)
 binomcdf(n, p, X) to find P(X <, <, >, > k)
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Sample Proportions Mean and St. Deviation
o μp-hat = p
o σp-hat = √((p*(1-p))/n)
o Using normal approximation
 p-hat is approximately normal when n*p and n*(1-p) must be
greater than or equal to 10
 normalcdf (lower bound, upper bound, μp-hat, σp-hat)
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Binomial probability
o P(X = k) = (nCk)(p)k(1-p)n-k
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Sample means
o x-bar
o μx-bar = μpop
o σx-bar = σ/√n
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Sampling Distribution of sample mean (x-bar)
o If the populations is normal with N(μ, σ) then the sample mean is N(μ,
σ/√n)
o If the population is not normal, then the sample mean is approximately
normal with N(μ, σ/√n) IF the sample size n is greater than or equal to
30.
You are NOT responsible for
- p. 385 through the top of page 387 (continuity correction, approximating X with
the normal distribution and with normalcdf)
- p. 406 through 407 (Weibull distributions)
- Section 5.3
Chapter 5 Review Packet
1. What are the 4 conditions of the Binomial Setting?
a.
b.
c.
d.
2. What is a sample proportion?
3. Circle the correct answers in each sentence
a. p is a (parameter/statistic) and it describes a (sample/population)
b. p-hat is a (parameter/statistic) and it describes a (sample/population)
c. x-bar is a (parameter/statistic) and it describes a (sample/population)
d. μ is a (parameter/statistic) and it describes a (sample/population)
4. I have a binomial distribution of a random variable X with B(300, 0.35).
a. What are n and p?
b. What is the mean and standard deviation of X?
5. What test must a binomial distribution pass in order to approximate it with the
normal distribution?
6. Finish the sentence:
Averages are _________________________ than individual observations
7. State the Central Limit Theorem in your own words
8. How does the Central Limit Theorem differ from the Law of Large Numbers?
9. What are the formulas for the mean and standard deviation of each of the
following:
a. Binomial random variable (X)
b. Sample proportion (p-hat)
c. Sample mean (x-bar)
10. I have a non-normal population, from which I want to take sample means. How
large must my sample size be in order to say that the distribution of my sample
mean is normal?
For the following problems, show all work. If you use calculator functions to
calculate probabilities, show what you put into the calculator.
11. Experts say that 44% of all American adults agree that parents should be given
vouchers for education at any public or private school of their choice. A national
opinion poll is taken for an SRS of 300 people. Assume that X (the number of
people that agree) is a binomial random variable
a. Is this a binomial distribution? Why?
b. What is the expected number of people that will agree?
c. What is the probability that exactly 175 people agree?
d. What is the probability that greater than 160 people agree?
e. What is the probability that 134 people or less agree?
f. What is the probability that 141 or more people agree?
g. What is the probability that less than 127 people agree?
12. A university (better known for its basketball program rather than its academics)
claims that 80% of its basketball players get degrees (within 5 years). An
investigation looks at the fate of 85 basketball players that entered the program
over a period of several years.
a. What is the expected (mean) percent of players that get degrees?
b. What is the probability that less than 75% of players get degrees?
c. What is the probability that more than 82% of players get degrees?
d. What is the probability that the percent of players that get degrees is
within 5% of the mean?
13. Billy takes a multiple-choice test. There are 115 questions. Billy knows some of
the material (but not much!), so his probability of getting a right answer is 0.57.
a. What is the expected number of questions that Billy will get right?
b. What is the expected score (percent) that Billy will get on the test?
c. What is the probability that Billy will get more than 85 questions right?
d. What is the probability that Billy will get between 65 and 80 questions
right?
e. What is the probability that Billy will get less than 63 questions right?
f. What is the probability that Billy will fail (failing is below a 60%)?
g. What is the probability that Billy will get between a 75% and an 85%?
h. What is the probability that Billy will get an A- or above (an A- is a 90%)
14. The number of accidents per week at a hazardous intersection varies with mean
2.2 and standard deviation 1.4. The distribution takes only whole number values,
so it is certainly not normal. Let x-bar be the mean number of accidents per week
at the intersection during a year (52 weeks).
a. What is the approximate distribution of x-bar (according to the Central
Limit Theorem)?
b. What is the probability that x-bar is less than 2?
c. What is the probability that there are fewer than 100 accidents at the
intersection in a year? (Hint: restate this event in terms of x-bar)
15. The scores on the SATs at a large high school are normally distributed with a
mean of 950 and a standard deviation of 85.
a. What is the probability that a randomly selected student from the high
school will have a score of 1000 or above?
b. What is the probability that a randomly selected student from the high
school will have a score of 1250?
c. What is the probability that a randomly selected student from the high
school will have a score of 800 or below?
d. If we take a sample of 100 students from the high school, what is the
distribution of x-bar, the mean of their scores on the SATs?
e. What is the probability that a randomly selected sample of 100 students
from the high school will have a mean score of 1200 or above?
f. What is the probability that a randomly selected sample of 100 students
from the high school will have a mean score of below 1050?
g. What mean score will have 75 % of the students below it?
16. We want to look at the number of hours of TV watched per day by each gender.
We are concentrating our efforts on working people ages 18-25. The distribution
of the number of hours watched per day is normally distributed for both genders.
The distribution for the men is N(2.25, 0.4), and for the women N(1.75, 0.5). The
hours watched by the men is independent of the hours watched by the women.
a. What is the distribution for the difference between the men and the women
(men – women)?
b. What is the probability that the difference between the men and the
women is more than 0.75 hours?
c. If I take a random sample of 20 men and 20 women, what is the sampling
distribution of the average difference between the men and the women?
d. What is the probability that the sample mentioned in part (c) has an
average difference of 0.75 or more?
ANSWERS
1. (a) fixed number of observations (n)
(b) probability of success (p) is the same for each observation
(c) all observations are independent
(d) only two possible outcomes, success or a failure
2. # of successes/total observations; X/n
3. (a) parameter; population
(b) statistic; sample
(c) statistic; sample
(d) population; parameter
4. (a) n = 300; p = 0.35
5. mean = 105; st. dev = 8.2614
6. “more normal and less variable”
7. As the sample size n increases, the sampling distribution of x-bar becomes more
normal
8. Law of Large numbers says that the overall average of n observations becomes
closer to the population mean as n increases. Central Limit Theorem says that as
the sample size n increases, the sampling distribution of the averages of repeated
samples becomes more normal
9. (a) mean = n*p; st. dev = sq.root(n*p*(1-p))
(b) mean = p; st. dev = sq.root(p*(1-p)/n)
(c) mean = µ; st. dev = σ/sq.root(n)
10. n must be greater than or equal to 30
11. (a) yes- see question #1
(b) expected = mean = n*p = 300*0.44 = 132
(c) binompdf(300, 0.44, 175) = 0.000000194
(d) 1 – binomcdf(300,0.44,160) = 0.000482
(e) binomcdf(300,0.44,134) = 0.6152
(f) 1-binomcdf(300,0.44,140) = 0.161406
(g) binompdf(300, 0.44, 126) = 0.261664
12. (a) mean = 0.8, st. dev = 0.043386
(b) check that n*p and n*(1-p) is greater than or equal to 10
0.1245695
(c) 0.3224
(d) 0.7509
13. (a) mean = n*p = 65.55
(b) mean = p = 0.57; st. dev = 0.046166
(c) 0.0000567
(d) binomcdf(115, 0.57, 85)- binomcdf(115, 0.57, 65) = 0.503147
(e) 0.2819
(f) check that n*p and n*(1-p) is greater than or equal to 10
0.742098
(g) 0.00004832
(h) 4.43 * 10^(-13) = almost 0
14. check to see that n is greater than or equal to 30  yes, 52 > 30
(a) x-bar is approximately N(2.2, 0.19415)
(b) 0.15147
(c) 0.0769
15. (a) 0.2782
(b) 0
(c) 0.03881
(d) mean = 950; st. dev = 8.5
(e) 0
(f) 1
(g) 955.733
16. (a) N(0.5, 0.6403)
(b) 0.3481
(c) N(0.5, 0.14318)
(d) 0.0404