Ch. 5 - Binomial Distribution
Ch. 6 - Normal Distributions
Ch. 7 - Sampling Distribution
Ch. 8 - Large Sample Estimation
General/Ch. 8
100
Which of the following examples is the random variable x binomial?
1) A box contains 6 blue candies and 5 green candies. A child randomly takes out 3 one by one without replacement. Let x denote the # of green candies selected.
2) A family in Cedar Falls is randomly chosen. Let x denote the # of males in the family.
Neither are binomial experiments
100
Find the following standard normal probabilities:
1) P(z < 3.85)
2) P(-1.7 < z < 2.8)
1) P(z < 3.85) ≈ 1
2) P(-1.7 < z < 2.8) = .9528
100
Which example(s) of the following satisfies the Central Limit Theorem?
1) A random sample size n = 25 from a normal distribution with μ = 25 and σ = 3.
2) A random sample size n = 150 from a normal distribution with μ = 50 and σ = 10.
3) A random sample size n = 25 from a binomial population with p = .4.
1) No - sample size < 30
2) Yes - sample size > 30
3) Yes - np = 25(.4) = 10 > 5 and nq = 25(.6) = 15 > 5
100
Double Points!!!
Solve for the z-score of the following confidence intervals:
1) 90% CI
2) 98% CI
1) 90% CI: 1.645
2) 98% CI: 2.33
100
Draw a number line for z-scores and label when a z-score is usual and unusual.
200
Toss a fair die 10 times and random variable x denotes the number of ones observed. Then x is a binomial random variable with:
1) n =
2) p =
1) n = 10
2) p = 1/6
200
Double Points!!!
Let z have a standard normal distribution. Find the following values of a:
1) P(z > a) = .85
2) P(-a < z < a) = .95
1) a = -1.04
2) a = 1.96
200
In a school system, the average ACT score is 23 and the standard deviation is 3 points. A sample of 100 students is randomly chosen from the system.
Find the mean and standard deviation of the sampling distribution.
Sample mean = population mean = 23
Sample SD = population SD/sqrt(n) = 3/sqrt(100) = .3
200
A quality control technician wants to estimate the proportion of soda cans that are underfilled. He randomly samples 175 cans of soda and finds 28 underfilled cans.
Calculate the point estimate and the margin of error.
Point estimate: p hat = .16
Margin of error: MOE = +/- .054
200
Explain the general process of finding a probability.
Such as: find the probability that x is greater than 10, with a given mean and standard deviation.
1) Standardize the interval: find the z-score using an equation
2) Use the z-score table to find the probability
300
Toss a fair die 10 times and random variable x denotes the number of threes observed.
Find the probability that we toss 3 threes.
P(x = 3) = 10C3 (1/6)^3 (5/6)^7 = 0.155
k = 3
n = 10
p = 1/6
q = 5/6
300
Random variable x has a normal distribution with μ = 8 and σ = 1.5. Find P(x > 9).
P(x > 9) = .2514
300
In a certain population, 15% of people have a disability. Let x denote the number of people who have the gene among the 500 selected people from the population.
What is the mean and standard deviation of the sample distribution.
Sample mean = p = .15
Sample SD = sqrt(pq/n) = sqrt(.15*.85/500) =.016
300
A university school president is interested in knowing what proportion of applicants would like to be accepted into the business department. Of a random sample of 68 applicants, 13 requested the business department.
Calculate the point estimate and margin of error.
Point estimate: p hat = .19
Margin of error: MOE = +/- .0932
300
Based on the following confidence intervals, which one(s) have a difference in the true mean or proportion?
1) -.01 < μ1 - μ2 < 2.58
2) -.7 +/- .35
3) -4.73 < p1 - p2 < -1.99
1) No (includes the possibility of μ1 - μ2 = 0)
2) Yes (simplifies to -1.05 < μ1 - μ2 < -.35)
3) Yes (does not include possibility of p1 - p2 = 0)
400
A quiz consists of 5 multiple-choice questions. Each question has 4 choices, with exactly one correct choice. A student, totally unprepared for the quiz, guesses on each question.
1) What is the expected # of questions they answer right?
2) What is the standard deviation of questions correct?
1) E(x) = np = 5(1/4) = 1.25
2) SD = sqrt(npq) = sqrt(5(1/4)(3/4)) = .968
400
Suppose that a particular species of trees have a normal distribution with an average diameter of 17 inches with a standard deviation of 2.5 inches.
Find the probability that a tree's diameter will be less than 20 inches.
P(x < 20) = .8849
400
In a market, the average money spent by each customer is $10 with a standard deviation of $3.5. Suppose that a random sample of 40 customers was selected.
What is the probability that the sample average is between $9.5 to $11.5
P(-.9 < z < 2.71) = .8120
400
A homeowner randomly samples 85 homes similar to her own and finds that the average selling price is $265,000 with a standard deviation of $12,000.
Find a 98% confidence interval for the average selling price.
261967.31 < μ < 268032.69
400
Compare the average daily intake of dairy products of men and women using a 90% confidence interval.
-16.724 < μ1 - μ2 < 4.724
500
A quiz consists of 5 multiple-choice questions. Each question has 4 choices, with exactly one correct choice. A student, totally unprepared for the quiz, guesses on each question.
What is the probability they answer at least 4 questions correctly?
P(x > 4) = P(4) + P(5) = .0156
500
Suppose that a particular species of trees have a normal distribution with an average diameter of 17 inches with a standard deviation of 2.5 inches.
Find the 90% percentile diameter.
x = 20.2 inches
500
Suppose a sample of 150 items is drawn from a population of manufactured products and the number of defective items is recorded. Prior experience has shown that the proportion of defective items is .1.
1) If we find that 20 items in this sample are defective, what is the observed value of p hat?
2) Do you think it is unusual?
1) p hat = x/n = 20/150 = .133
2) z = 1.347, Not unusual
500
A random sample of n = 100 males showed a mean average daily intake of dairy products equal to 746 grams with a standard deviation of 30 grams.
Find a 90% CI for the population average μ, true average daily intake.
741.065 < μ < 750.935
500
In a study comparing medicines, 190 randomly selected people used medicine A, with 167 having positive results, and 235 randomly selected people used medicine B, with 212 having positive results.
Find a 95% confidence interval for the difference in the true proportions experiencing positive results from the two medicines.
-.08 < p1 - p2 < .04