Statistics Ch. 5 & 6
Probabilities are always numbers between and including what numbers?
0 and 1
In most cases, it is recommended that at least how many trials be done when using a simulation to estimate a probability?
100
What does a probability distribution indicate?
All the possible outcomes of a random experiment and the probability of each outcome of a random experiment
Which of the following cannot be used to represent a discrete probability distribution?
Areas under curves
Which of the following cannot be used to represent a discrete probability distribution?
Areas under curves
The normal model is a good first choice to model data if the data are suspected to be ________.
Symmetric and unimodal
Often, conditional probabilities are worded with what phrase?
"given that"
Suppose that the probability that a person books an airline ticket using an online travel website is 0.72. Consider a sample of ten randomly selected people who recently booked an airline ticket. What is the probability that no more than three out of ten people used an online travel website when they book their airline ticket? Round to the nearest thousandth.
0.007
Suppose that the probability that a person books a hotel using an online travel website is 0.68. For the questions that follow, consider a sample of fifteen randomly selected people who recently booked a hotel. What is the probability that at least fourteen out of fifteen people used an online travel website when they booked their hotel? Round to the nearest thousandth.
0.025
Suppose that the probability that a person books an airline ticket using an online travel website is 0.72. Consider a sample of ten randomly selected people who recently booked an airline ticket. What is the probability that at least nine out of ten people used an online travel website when they booked their airline ticket? Round to the nearest thousandth.
0.183
Determine whether the variable would best be modeled as continuous or discrete: The temperature of a greenhouse at a certain time of day.
Continuous
What are numerical variables with outcomes that cannot be listed or counted because they occur over a range called?
Continuous
Determine whether the variable would best be modeled as continuous or discrete: The number of tomatoes harvested each week from a greenhouse tomato plant
Discrete
Probabilities that are based on short-run relative frequencies are called what?
Empirical probabilities
A casino claims that its roulette wheel is truly random. What should that claim mean?
Every number is equally likely to occur.
The notation P(F|E) means the probability of event _____ given event _____.
F, E
Theoretical probabilities are:
the relative frequencies at which an event happens after infinitely many repetitions
The total area under a probability density curve ______.
equals 1
The total area under a probability density curve _____.
equals 1
Often, conditional probabilities are worded with what phrase?
"given that"
Determine whether the random variable is discrete or continuous. In each case, state the possible values of the random variable. (a) The number of fish caught during a fishing tournament (b) The amount of rain in City during April. (c) The amount of snowfall
(a) The random variable is discrete. The possible values are x = 0, 1, 2,... (b) The random variable is continuous. The possible values are r≥0. (c) The random variable is continuous. The possible values are s≥0.
To generate random numbers, one can:
- Use a random number table - Use computer program - Use the internet
Suppose that the probability that a person books a hotel using an online travel website is 0.68. Consider a sample of fifteen randomly selected people who recently booked a hotel. What is the probability that exactly ten people out of fifteen people used an online travel website when they booked their hotel? Round to the nearest thousandth.
0.213
A single die is rolled. Find the probability of rolling an even number or a number less than 6.
1
Which of the following is the probability that something in the sample space will occur?
1
Suppose that the probability that a person books a hotel using an online travel website is 0.68. Consider a sample of fifteen randomly selected people who recently booked a hotel. Out of fifteen randomly selected people, how many can be expected to use an online travel website to book their hotel, give or take how many? Round to the nearest whole person.
10 people, give or take 2 people
The percentage of left-handed people in a certain country is estimated to be 9%. Women are about six times as likely to be left-handed as men. Are gender and handedness independent or associated? Explain.
Gender and handedness are associated because women are more likely to be left-handed than men.
A 2008 survey found that 48% of women in a certain country agreed that same-sex marriage should be allowed, but only 37% of men felt that way. Suppose that these figures are accurate (or nearly accurate) probabilities, and state whether gender and opinion about same-sex marriage are independent. Explain.
Gender and opinion about same-sex marriage are not independent; they are associated. The women were more likely to support same-sex marriage.
Which of the following is the best explanation to what should happen to the proportion of heads as the number of coin flips increases?
Gets closer to 0.5
Variables or events that are not associated are called what?
Independent
In a simulation of coin tosses, a streak of 15 heads has appeared. The Law of Large Numbers says which of the following must be true?
It is equally likely that the 16th toss will be a head or a tail.
When events A and B are said to be independent, what does that mean?
Knowledge that event B occurred doesn't change the probability of event A occurring.
If an experiment with a random outcome is repeated a large number of times, the empirical probability of an event is likely to be close to the true probability. This mathematical theorem is called what?
Law of Large Numbers
If events A and B are independent, what must be done to find the probability of event A AND B?
Multiply the probability of A and the probability of B.
When two events have no outcomes in common, they are called what?
Mutually exclusive
Decide if the events are mutually exclusive. Event A: Randomly selecting someone who owns a car Event B: Randomly selecting a married male
No, because someone who owns a car can be a married male.
If 23% of Americans households own one or more dogs and 42% own one or more cats, then from this information, is it possible to find the percentage of households that own a cat OR a dog? Why or why not?
No, because the event of owning a dog and the event of owning a cat are not mutually exclusive. Therefore, to find the percentage of people that own a cat or a dog, it is necessary to know the percentage of people that own a cat and a dog.
Determine whether the following probability experiment represents a binomial experiment and explain the reason for your answer. Three cards are selected from a standard 52-card deck without replacement. The number of sevens selected is recorded
No, because the trials of the experiment are not independent and the probability of success differs from trial to trial.
Determine whether the following probability experiment represents a binomial experiment and explain the reason for your answer. Three cards are selected from a standard 52-card deck without replacement. The number of sevens selected is recorded.
No, because the trials of the experiment are not independent and the probability of success differs from trial to trial.
Statistics and probability use the "inclusive OR". This means that referring to outcomes A OR B is referring to what?
Outcomes that are only in A, only in B, or in both
Imagine flipping a fair coin many times. Explain what should happen to the proportion of heads as the number of coin flips increases.
Proportion should get closer to 0.5
Because they are generated by a seed value that starts the random sequence, computer-generated random numbers are sometimes called what?
Pseudo-random numbers
In statistics, what is true of randomness?
Randomness is hard to achieve without help from a computer or some other randomizing device.
Experiments used to produce empirical probabilities are called what?
Simulations
Law of large numbers
States that if an experiment with a random outcome is repeated a large number of times, the empirical probability of an event is likely to be close to the true probability. The larger the number of repetitions, the closer together these probabilities are likely to be.
Given the event "a die lands with a 6 on top", which of the following is the complement of this event?
The die lands with a 1, 2, 3, 4, or 5 on top
When people fold their hands together with interlocking fingers, most people are more comfortable with one of two ways. In one way, the right thumb ends up on top and in the other way, the left thumb is on top. The table shows the data from one group of people. Judging on the basis of this data set, are the events "right thumb on top" and "male" independent or associated?
The events "right thumb on top" and "male" are independent because knowing that a subject prefers to put their right thumb on top does not change the probability that they are male.
What is the most widely used probability model for continuous numerical variables?
The normal distribution
A teacher wants to find out whether the chance of drawing an Ace is 7.7 %. In the last 5 minutes of class, he has all the students draw cards, replacing the previous card and shuffling between each draw, until the end of class and then report their results to him. Which condition or conditions for use of the binomial model is or are not met?
The number of trials is fixed
Which of the following characteristics are not required for the binomial model?
The outcome must be a discrete numerical variable
Which of the following characteristics are not required for the binomial model?
The probability of success must be the same as the probability of failure.
The sample space of a random experiment is what?
The set of all possible and equally likely outcomes of the experiment
What is true of the shape of a binomial distribution?
The shape depends on both the number of trials, n, and the probability of success, p
What are the values of the mean and the standard deviation for the standard normal model?
The standard normal model has a mean of 0 and a standard deviation of 1
Which of the following might be a reason that an empirical probability found through a simulation does not match the theoretical probability?
The theoretical value is incorrect The empirical value is just varying. Doing more trials will get closer to the theoretical probability.
Which of the following is not a characteristic that must be present in a binomial model?
The trials are dependent.
Suppose that a person who has recently helped start a band for the first time is randomly selected. The probability that the band will break up within 4 months is 0.5. Suppose we follow 11 bands (44) people for 4 months and record the number of people whose band broke up. Why is the binomial model inappropriate for finding the probability that at least 6 of these 44 people will no longer be in that band within 4 months? List all binomial conditions that are not met.
The trials are independent.
What determines the exact shape of a normal distribution?
The values of the mean and the standard deviation
A friend flips a coin 30 times and says that the probability of getting a head is 40% because he got twelve heads. Is the friend referring to an empirical probability or a theoretical probability? Explain.
This is an example of empirical probability because it is based on an experiment.
A Monopoly player claims that the probability of getting a 5 when rolling a six-sided die is one sixth 1/6 because the die is equally likely to land on any of the six sides. Is this an example of a theoretical probability or an empirical probability? Explain.
This is an example of theoretical probability because it is not based on an experiment.
True or False. Random means that no predictable pattern occurs and that no digit is more likely to appear than any other.
True
Determine whether the following probability experiment represents a binomial experiment and explain the reason for your answer. An experimental drug is administered to 100 randomly selected individuals, with the number of individuals responding favorably recorded
Yes, because the experiment satisfies all the criteria for a binomial experiment
Determine whether the following probability experiment represents a binomial experiment and explain the reason for your answer. An experimental drug is administered to 100 randomly selected individuals, with the number of individuals responding favorably recorded.
Yes, because the experiment satisfies all the criteria for a binomial experiment.
A married couple plans to have four children, and they are wondering how many boys they should expect to have. Assume none of the children will be twins or other multiple births. Also assume the probability that a child will be a boy is 0.50. Explain why this is a binomial experiment. Check all four required conditions.
Yes, there are two complementary outcomes, either a boy or a girl. Yes, there are 4 fixed trials, since there are 4 children. Yes, the probability of having a boy is 0.50 for each child. Yes, because it is assumed there are no twins, the gender of one child does not affect the gender of another.
Consider the following categories of people, assuming that we are talking about all the people in a certain country. Category 1: People who have a job Category 2: People who are in school Category 3: People who have a job OR are in school Category 4: People who have a job AND are in school a. Which of the four categories has the most people? b. Which category has the fewest people?
a. Category 3 b. Category 4
In statistics, for what does the abbreviation "pdf" stand?
probability distribution function
What is another name for the expected value of a probability distribution?
the mean
The binomial probability model is useful in many situations with variables of what kind?
Discrete-valued numerical variables
What are the two requirements for a discrete probability distribution?
∑P(x)=1 and 0≤P(x)≤1