Stats Exam 3
Suppose an aircraft has two warning systems that sound an alert if the craft is approaching terrain (a mountain). In the presence of terrain, system A sounds 90% of the time. If system A fails, system B is called. System B sounds 80% of the tme in the presence of terrain. In the presence of terrain, find the probability an alert will be sounded.Hint: Draw a tree diagram -- first branching is first alarm, second set of branches is second alarm.
(0.9) + (0.1)(0.8) = 0.98
If the odds are 8 to 1 against a team winning (they lose more than they win), what is the probability the team wins?
1/9
Odds are 11 to 2 the horse will win the race. What is the probability the horse will win?
11/13
a sorority is selling $1 raffle tickets (1/1000) chance of winning $200, expected gain?
199 (1/1000) + -1(999/1000)
Odds are 2 the horse will win the race. What is the probability the horse will win?
2/3
For a 95% confidence interval, which of the following is a true statement?
95% of the intervals in the long run will contain the parameter
A basketball player makes 65% of her shots from the field during the season. To simulate whether a shot hits or misses which one of the following would be an appropriate assignment of digits? A) One digit simulates one shot; 6 and 5 are a hit, other digits are a miss. B) One digit simulates one shot; odd digits are a hit and even digits are a miss. C) Two digits simulate one shot; 01 to 65 are a hit and 66 to 99 are a miss. D) Two digits simulate one shot; 01 to 65 are a hit and 66 to 99 and 00 are a miss.
D
The psychologist Amos Tversky did various studies of our perception of chance behavior. In its obituary of Tversky, the New York Times cited the following example. Tversky asked subjects to choose between two public health programs that affect 600 people. The first program has probability 1/2 of saving all 600 and probability 1/2 that all 600 will die. As the second program, the subjects were given two options, Option A and Option B. Option A is guaranteed to save exactly 400 of the 600 people. Option B will definitely lose exactly 200 lives. Identify the correct statement with reference to Option A and Option B of the second program.
Both options are equivalent because they save an equal number of lives
The probability of getting a speeding ticket today is 0.05. The probability of winning a scratch off lottery you purchased today is 0.10. Assuming that getting a speeding ticket has nothing to do with whether your ticket is the winning one or not, find the probability that you get a speeding ticket and win the lottery
0.05 * 0.10 = 0.005
The probability it will rain on Wednesday morning is 0.3. The probability it will rain Wednesday afternoon is 0.3. The probability it will rain both in the morning and the afternoon is 0.10. Find the probability it will rain Wednesday morning or afternoon
0.3 + 0.3 - 0.1 = 0.5
A Gallup poll on Presidents Day 2008 interviewed a random sample of 1007 adult Americans. They were asked which former president they would like to bring back as the next president if they could. Here are the results: Outcome Probability John F. Kennedy 0.23 Ronald Reagan 0.22 Abraham Lincoln 0.10 Someone else ? These proportions are probabilities for the random phenomenon of choosing an adult American at random and asking her or his opinion. What must be the probability that the person chosen selects someone other than John F. Kennedy, Ronald Reagan, or Abraham Lincoln?
0.45
Government data assign a single cause for each death that occurs in the United States. The data show that the probability is 0.27 that a randomly chosen death was due to heart disease, and is 0.23 that it was due to cancer. What is the probability that a death was due to either heart disease or cancer?
0.50
Choose an acre of land in Canada at random. The probability is 0.45 that it is forest and is 0.03 that it is pasture. What is the probability that the chosen acre is not forested?
0.55
In government data, a family consists of two or more persons who live together and are related by blood or marriage. Choose an American family at random and count the number of people it contains. Here is the assignment of probabilities for your outcome: Number of persons 2 3 4 5 6 7+ Probability 0.42 0.23 0.21 0.09 0.03 0.02 What is the probability that the family you choose has more than two people?
0.58
Consider the experiment where you roll a fair coin twice and observe the number of Tails. The probability distribution is: #T prob 0 0.25 1 0.5 2 0.25 What is the complement of 0 Tails?
1 or 2 tails
Larry Bird made 90% of his free throws. To simulate one free throw shot by Larry Bird, we could use a random digit with A) odd = made, even = missed. B) 0 to 8 = made, 9 = missed. C) 0 to 4 = made, 5 to 9 = missed. D) 1 to 9 = made, 0 = missed. Either (A) or (C) is correct. Either (B) or (D) is correct.
Either b or d
Suppose you conduct a hypothesis test and you reject the null hypothesis (HO) and find in favor of the alternative hypothesis (HA). You can conclude
Either the null or alt could be true
Suppose the probability of a horse winning a race is 0.85. What is the complement of the horse winning?
Horse not winning
When individual outcomes are uncertain but there is a regular distribution of outcomes in a large number of repetitions it is _____
Random
if the p value is less than the or equal to the level of significance then we must
Reject the null
A lottery has a game where 1 ball is drawn from each of 4 bins. Inside each bin are 10 balls, numbered 0 through 9. Which outcome has a higher probability of occurring? 3576 or 0000
They both have the same probability of occurring
A basketball player makes 65% of her shots from the field during the season. To simulate whether a shot hits or misses which one of the following would be an appropriate assignment of digits?
Two digits simulate one shot; 01 to 65 are a hit and 66 to 99 and 00 are a miss.
P(A or B) means to
add; it is disjoint
the _____ is the hypothesis we are looking for evidence for
alternative/ research hypothesis
the sampling distribution of a sum or percentage will become approximately normal as the sample size gets larger
central limit theorem
the population mean uses the _____ to assume the sampling distribution of the sample mean is normal (with a large enough sample size)
clt
two events have no outcomes in common
disjoint
Sample space is all the things you could do today. There is a 0.20 chance you will go boating all day. There is a 0.10 chance you will go shopping all day. There is a 0.7 chance you will do something else all day. We would say the events "shop all day" and "boat all day" are
disjoint and dependent
the smaller the p value, the more ____ against the null hypothesis
evidence
the ______ of a phenomenon that has numerical outcomes is found by multiplying each outcome by its probability and then adding all of the products
expected value
If we increase the sample size we'll get an interval that is
narrower
the bigger the sample size the _____ the confidence interval
narrower
the _____ is the probability that we would see a statistic at least as extreme as the one observed if the null hypothesis was true
p value
A ____ is a number used to describe a population
parameter
In statistics, _____ describes an order that would only occur in the long run
random
collection of unique outcomes of a random circumstance
sample space
different samples from the same population may yield different values of the sample statistic is called
sampling variability
a small sample can miss an important difference by being more _____
sensitive (reject less often)
In the law of large numbers, repetitions of a random phenomenon are likely to become more ____ as the number of trials increases
stable
A sampling distribution gives the distribution of the values assumed by the
statistic
the p value determines the ____ _____
statistical significance
a _____ is a number used to describe a sample
statistics
The probability of a random outcome is
the proportion of times the outcome occurs in a large number of repetitions
if we make some assumptions based on a set of theories we can calculate the probability based on the theories
theoretical
The Law of Large Numbers can be interpreted to mean that
you can count on approximately 50% of the flips of a fair coin to be Heads in a large number of flips and the larger the number of flips, the better this approximation
3 reasons why there is a difference between what we consider risky and what experts consider risky
we feel safer when risk is under control, hard to comprehend small probabilities, experts use complicated stat models
The higher level of confidence the ______ the confidence interval
wider
a large sample size makes a test of significance more ___-
sensitive (reject more often)
Odds in terms of 1 number: #
#/ (#+ 1)
The local convenience store sells scratch off tickets for $1.00 each. The probabilities and amounts to be won are printed on the ticket as follows: Amount won Probability $2000 0.0001 $50 0.01 Find the expected gain for a customer who plays.
$ -0.30
A grocery chain runs a prize game by giving each customer a ticket that may win a prize when a box is scratched. Printed on the ticket are the following probabilities for a customer who shops once a week: Amount won Probability $1000 0.01 $100 0.10 $10 0.20 What is the expected value of a customer's winnings in this game?
$22
The probability of being left handed is 0.088, the probability of being a female is 0.73, and the probability of being a female and left-handed is 0.073. Find the probability of a randomly selected person from this class being left-handed OR a female.
0.088 + 0.73 - 0.073 = 0.745
Find the probability of rolling an even number on the first and second roll of a fair six-sided die.
1/4
A baseball player makes 30% of his at-bats during the season. Use 00 to 29 as a hit and 30 to 99 are a miss. Using that information, use these random digits to simulate 10 at-bats: 82234 71490 20467 47511 81676 55300 94383 14893 How many of these 10 at-bats are hits?
4
How many possible outcomes are there when rolling 3 fair dice?
6*6*6 = 216
Odds in terms of 2 numbers: A to B
A / (A plus B)
How can we make a confidence interval narrower?
Increase the sample size
If a random phenomenon with numerical outcomes is repeated many times independently, the mean of the observed outcomes (sample mean) approaches the expected value
Law of large numbers
A number between 0 and 1 that describes the proportion of times an outcome would occur in a very long series of repetitions is called _____
Probability
_____ Describes the long term irregularity of events
Probability
the ______ of a statistic is the distribution of values taken by the statistic in all possible samples from the same population
sampling distribution
Based upon the results of a hypothesis test, researchers find in favor of the alternative hypothesis. This means their test resulted in
a small p value
to be legitimate, it has to
add up to one and be between 1 and 0
We repeat an experiment many times and calculate the proportion of time each experiment occurs
empirical
A random outcome is _____ in the short run but _____ in the long run
not predictable; predictable
the _____ is what we have no choice but to believe until we find evidence otherwise
null hypothesis; always gets the equal sign
Historically, 76% of students in an introductory psychology course have correctly answered the professor's favorite test question on Freud. This semester the professor gave a randomly selected group of students an extra lecture on the subject and wants to see if it will help them do better on the question. The null hypothesis (HA) should be:
p > .76
how to find the critical value of a confidence interval
p hat + Z( sd of distribution)
The _____ is used to summarize the amount of evidence we have against the null hypothesis
p value
how to solve a union
p(a) + p(b)- p(a and b)
A grocery chain runs a prize game by giving each customer a ticket that may win a prize when a box is scratched. Printed on the ticket are the following probabilities for a customer who shops once a week: Amount won Probability $1000 0.01 $100 0.10 $10 0.20 If several thousand customers play the grocery store game, you expect that the mean amount they win will be close to
the expected value of the customer's winnings
lack of evidence for the alternative does not prove that
the null is true
standard deviation of the distribution
sqaure root p (1-p)/ n
why are confidence intervals more informative than hypothesis test?
they estimate the parameter and are easier to understand
the ____ checks sample data against a claim or assumption about the population
hypothesis test
Odds range from 0 to ___
infinity
If a random phenomenon with numerical outcomes is repeated many times independently, the mean of the observed outcomes approaches the expected values
law of large numbers
the ______ determines how much evidence against the null we require to reject the null
level of significance
Suppose the probability of a horse winning a race is 0.85. What is the probability of the complement of the horse winning the race?
0.15
In the previous question, you simulated 10 at bats and recorded the number that were hits. Now suppose you repeat this simulation (of 10 at-bts) 20 times and get the following results: 2 4 3 2 5 1 6 4 1 3 2 5 4 8 4 3 1 6 2 3 Note that the first "2" represents a simulation of 10 at-bats resulting in 2 hits. Each number represents the results of a simulation of 10 at-bats. Use the results of the 20 repetitions to estimate the proability of getting three or fewer hits on any given 10 at-bats.
11/20
Using random digits from a table or from computer software to imitate choice behavior is called
Simulation
The P-value of a hypothesis test tells us
how often we would see the result if the null were true
The psychologist Amos Tversky did various studies of our perception of chance behavior. In its obituary of Tversky, the New York Times cited the following example. Tversky asked subjects to choose between two public health programs that affect 600 people. The first program has probability 1/2 of saving all 600 and probability 1/2 that all 600 will die. As the second program, the subjects were given two options, Option A and Option B. Option A is guaranteed to save exactly 400 of the 600 people. Option B will definitely lose exactly 200 lives. Fill in the blank: The expected number of people saved by the first program is ____
300 people
______ requires a large enough sample size to approximate the distribution of p hat with a normal distribution
CLT
the _____ describes all possible outcomes and says how to align probabilities
probability model
the reader determines ____ while the p value determines _____
practical significance and statistical significance
the reader can determine whether the result is
practically significant
our own personal judgmenet
personal probability
collection of outcomes
event