Statistics and Probability Answers

A bag contains 5 red 6 green 4 blue and 2 white beads; 6 beads are drawn at random from the bag what is the probability that:

a) 3 are red 2 or green and one is blue?
b) 2 are green 2 are blue and 2 are white?
c) 3 are blue
d) At least 4 are green
e) At least 1 is red bead?

A box contains 500 envelopes, of which 75 contain $100 in cash, 150 contain $25, and 275
contain $10. An envelope may be purchased for $25. What is the sample space for the different amounts
of money? Assign probabilities to the sample points and then find the probability that the first envelope
purchased contains less than $100

1. We would like to test the following hypotheses

H0 :p = 0.05
HA :p < 0.05

The sample size is 150 and the sample proportion is 8%, i.e. p hat = 0.08. Which of the below is the correct test for this situation?

A. z-test for a proportion
B. t-test for a mean
C. z-test for comparing two proportions
D. chi-square test of independence
E. randomization test for a proportion

2. At a stop sign, some drivers come to a full stop, some come to a ‘rolling stop’ (not a full stop, but slow down), and some do not stop at all. We would like to test if there is an association between gender and type of stop (full, rolling, or no stop). We collect data by standing a few feet from a stop sign and taking note of type of stop and the gender of the driver. What are the hypotheses for testing for an association between gender and type of stop?

A. H0: Males and females are equally likely to come to a rolling stop.
HA: Males are more likely than females to come to a rolling stop.

B. H0: Gender and type of stop are associated.
HA: Gender and type of stop are independent.

C. H0: Males and females are equally likely to come to a full stop.
HA: Males and females are not equally likely to come to a full stop.

D. H0: Gender and type of stop are independent.
HA: Gender and type of stop are associated.

1. 80% of Americans start the day with a cereal breakfast. Based on this information, determine if the following statement is true or false.

“The sampling distribution of the proportions of Americans who start the day with a cereal breakfast in random samples of size 40 is right skewed.”

A. True
B. False

A report on TV news station claims that 60% of the city’s residents support using limited city funds to hire and train more police officers. A second local news station has picked up this. The second news station takes a random sample of 100 residents and finds that 56 of them support the use of limited funds to hire additional police officers. Find the correct set-up for calculating the p-value for this test?

A. Randomly sample 100 residents of a nearby city, and record the number of residents in the sample who support the hiring of additional police officers. Repeat this many times and calculate the proportion of samples where at least 56% of the residents support the hiring.

B. Roll a 10-sided die 100 times and record the proportion of times you get a 6 or lower. Repeat this many times, and calculate the number of simulations where the sample proportion is 56% or less.

2. The campaign manager for a congressional candidate claims that the candidate has more than 50% support from the district’s electorate. A newspaper collects a simple random sample of 500 likely voters in this district and estimates the support for this candidate to be 52%. The p-value for the hypothesis test evaluating the campaign manager’s claim is 0.19. Which of the below is correct?

A. The success-failure condition is not met, so this p-value is not reliable.
B. 95% of random samples of size 500 will estimate the support for this candidate to be 52%.
C. If in fact 50% of likely voters support this candidate, the probability of obtaining a random sample of 500 likely voters where 52% or more support the candidate is 0.19.
D. The data provide convincing evidence for the campaign manager’s claim.

1. In May 2011, Gallup asked 1,721 students in grades five through twelve if their school teaches them about money and banking. Researchers are interested in finding out if a majority of students receive such education. Which of the following is the correct set of hypotheses?

A. H0 :μ = 0.5; HA :μ > 0.5
B. H0 :p < 0.5; HA :p > 0.5
C.H0 :p = 0.5; HA :p > 0.5
D. H0: hat p = 0.5; HA: hat p is not equal to 0.5

1. Suppose you want to construct a confidence interval for a population proportion. Which of the following, if it were true, would prevent you from being able to assume that the distribution of the sample proportion is nearly normal?

A. n = 104. Out of these 104 there are an equal number of successes and failures (52 each).
B. n = 104. Out of these 104 there are only a few successes (15), but relatively many failures (89).
C. None of these options.
D. n = 104. These observations are a simple random sample and make up less than 10% of the population.


2. When performing a hypothesis test on proportions (either where H0: p = p0 or where H0: p1 = p2) you should use the observed number of successes and failures when checking conditions.

A. True
B. False
C. Depends on the context
D. Observed proportion for one sample, expected proportion for two samples.

data was collected from a random sample of 330 home sales from a community in 2003. Let Price denote the selling price (in $1,000), BDR denote the number of bedrooms, Bath denote the number of bathrooms, Hsize denote the size of the house (in square feet), Lsize denote the lot size (in square feet), Age denote the age of the house (in years), and Poor denote a binary variable that is equal to 1 if the condition of the house is reported as "poor."
An estimated regression yields: Price with caret equals 108.6 plus 0.561 BDR plus 26.6 Bath plus 0.237 Hsize plus 0.002 Lsize
plus 0.1 Age minus 56.3 Poor comma Upper R overbar squared equals 0.84 comma SER equals 45.3.
Suppose that a homeowner converts part of an existing family room in her house into a new bathroom. What is the expected increase in the value of the house?
The expected increase in the value of the house is $
nothing. (Round your response to the nearest dollar.)