Statistical Patterns and Random Phenomena - AP Statistics
Card 1 of 560
Students collected 150 cans for a food drive. There were 23 cans of corn, 48 cans of beans, and 12 cans of tomato sauce. If a student randomly selects one can to give away, what is the probability that the can will be either tomato sauce or beans?
Students collected 150 cans for a food drive. There were 23 cans of corn, 48 cans of beans, and 12 cans of tomato sauce. If a student randomly selects one can to give away, what is the probability that the can will be either tomato sauce or beans?
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In this case, we want to know the probability of multiple, mutually exclusive possible outcomes. The possible outcomes are mutually exclusive because one can of food could not be both beans and tomato sauce. To determine the probability of the two possible outcomes, add them together and then find the least common denominator.
In this case, we want to know the probability of multiple, mutually exclusive possible outcomes. The possible outcomes are mutually exclusive because one can of food could not be both beans and tomato sauce. To determine the probability of the two possible outcomes, add them together and then find the least common denominator.
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150 students are athletes at the school. 65 play baseball, 15 play basketball, and 10 play both basketball and baseball.
What is the probability that a randomly selected athlete will play either baseball or basketball or both sports?
150 students are athletes at the school. 65 play baseball, 15 play basketball, and 10 play both basketball and baseball.
What is the probability that a randomly selected athlete will play either baseball or basketball or both sports?
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We want to know the probability of multiple possible outcomes that are not mutually exclusive. To do this, we use the addition rule with one step that we would not use if the possible outcomes were mutually exclusive. Add the probabilities of each possible outcome, subtract from that sum the number counted twice, then reduce the answer to the least common denominator.
We want to know the probability of multiple possible outcomes that are not mutually exclusive. To do this, we use the addition rule with one step that we would not use if the possible outcomes were mutually exclusive. Add the probabilities of each possible outcome, subtract from that sum the number counted twice, then reduce the answer to the least common denominator.
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Assume there is an election involving three parties: D, R, and I. The probability of D winning is .11, R winning is .78, and I winning is .11. What is the probability of D or R winning?
Assume there is an election involving three parties: D, R, and I. The probability of D winning is .11, R winning is .78, and I winning is .11. What is the probability of D or R winning?
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Since all of the events are mutually exclusive (one of the parties must win), you can get the probability of either D or R winning by adding their probabilities.
Since the probability of D winning is .11 and R winning is .78, the probability of D or R winning is .89.
Since all of the events are mutually exclusive (one of the parties must win), you can get the probability of either D or R winning by adding their probabilities.
Since the probability of D winning is .11 and R winning is .78, the probability of D or R winning is .89.
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If 1 card is chosen at random from a deck of cards, what is the probability that it will be a heart or a king?
If 1 card is chosen at random from a deck of cards, what is the probability that it will be a heart or a king?
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In a deck of cards, there are 52 total cards, 13 hearts, 4 kings, and 1 king that is a heart.
So, 
In a deck of cards, there are 52 total cards, 13 hearts, 4 kings, and 1 king that is a heart.
So,
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Using a standard deck of cards, what is the probability of choosing a single face card?
Using a standard deck of cards, what is the probability of choosing a single face card?
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There are 
cards in a standard deck: 
cards in 
suits. There are 
face cards (King, Queen, Jack) in each suit, so there are 
total face cards.
Thus the probability of choosing a single face card is 
or 
.
There are
Thus the probability of choosing a single face card is
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A student randomly selected a highlighter from her desk. There were five highlighters on the desk, each of a different color--blue, green, yellow, red, and orange. What is the probability that the student selected either the red or the yellow highlighter?
A student randomly selected a highlighter from her desk. There were five highlighters on the desk, each of a different color--blue, green, yellow, red, and orange. What is the probability that the student selected either the red or the yellow highlighter?
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In this case, we want to know the probability of multiple, mutually exclusive possible outcomes. To determine the probability of the two possible outcomes, simply add them together. This is called the addition rule.
In this case, we want to know the probability of multiple, mutually exclusive possible outcomes. To determine the probability of the two possible outcomes, simply add them together. This is called the addition rule.
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Which of the following populations has a precisely normal distribution?
Which of the following populations has a precisely normal distribution?
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A normal distribution is one in which the values are evenly distributed both above and below the mean. A population has a precisely normal distribution if the mean, mode, and median are all equal. For the population of 3,4,5,5,5,6,7, the mean, mode, and median are all 5.
A normal distribution is one in which the values are evenly distributed both above and below the mean. A population has a precisely normal distribution if the mean, mode, and median are all equal. For the population of 3,4,5,5,5,6,7, the mean, mode, and median are all 5.
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If a population has a normal distribution, the number of values within one positive standard deviation of the mean will be . . .
If a population has a normal distribution, the number of values within one positive standard deviation of the mean will be . . .
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In a normal distribution, the number of values within one positive standard deviation of the mean is equal to the number of values within one negative standard deviation of the mean. The reason for this is that the values below the population mean exactly parallel the values above the mean.
In a normal distribution, the number of values within one positive standard deviation of the mean is equal to the number of values within one negative standard deviation of the mean. The reason for this is that the values below the population mean exactly parallel the values above the mean.
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Which is the following is true about the standard normal distribution?
Which is the following is true about the standard normal distribution?
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The standard normal distribution is just like any other normal distribution that you might have looked at except that it has a standard deviation of 1 and a mean of 0.
The standard normal distribution is just like any other normal distribution that you might have looked at except that it has a standard deviation of 1 and a mean of 0.
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Which of the following is NOT a discrete random variable?
Which of the following is NOT a discrete random variable?
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By definition, a discrete random variable is a random variable whose values can be "counted" one by one. A continuous random variable is a random variable that can take any value on a certain interval. Of these choices, the number of lip products, the amount of money, and the number of midterms taken are all discrete random variables, as the respective values can be counted; however, the time taken to watch the first four seasons of a TV show is a continuous random variable, as not everyone will take the same amount of time to watch all those episodes (i.e. some might fastf-orward/replay parts of episodes).
By definition, a discrete random variable is a random variable whose values can be "counted" one by one. A continuous random variable is a random variable that can take any value on a certain interval. Of these choices, the number of lip products, the amount of money, and the number of midterms taken are all discrete random variables, as the respective values can be counted; however, the time taken to watch the first four seasons of a TV show is a continuous random variable, as not everyone will take the same amount of time to watch all those episodes (i.e. some might fastf-orward/replay parts of episodes).
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Cheyenne is worried about food thieves in the break room at work, and she believes that, as the week progresses, and people get lazy and ready for the weekend, more food theft occurs. She gathered the following data on number of thefts per day, and fell very behind in her work for a week.
Which of the following statements about the data are true?
i: the data is normally distributed
ii: the data is skewed left
iii: the data supports Cheyenne's theory
iv: the data is a representative sample
Cheyenne is worried about food thieves in the break room at work, and she believes that, as the week progresses, and people get lazy and ready for the weekend, more food theft occurs. She gathered the following data on number of thefts per day, and fell very behind in her work for a week.
Which of the following statements about the data are true?
i: the data is normally distributed
ii: the data is skewed left
iii: the data supports Cheyenne's theory
iv: the data is a representative sample
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The data is not normal by virtue of being skewed left, which also supports Cheyenne's theory... there is no way of knowing wether this data was a representative sample, but also no option with ii, iii and iv was provided to avoid frustration/confusion
The data is not normal by virtue of being skewed left, which also supports Cheyenne's theory... there is no way of knowing wether this data was a representative sample, but also no option with ii, iii and iv was provided to avoid frustration/confusion
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Alex took a test in physics and scored a 35. The class average was 27 and the standard deviation was 5.
Noah took a chemistry test and scored an 82. The class average was 70 and the standard deviation was 8.
Show that Alex had the better performance by calculating -
-
Alex's standard normal percentile and
-
Noah's standard normal percentile
Alex took a test in physics and scored a 35. The class average was 27 and the standard deviation was 5.
Noah took a chemistry test and scored an 82. The class average was 70 and the standard deviation was 8.
Show that Alex had the better performance by calculating -
-
Alex's standard normal percentile and
-
Noah's standard normal percentile
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Alex -
on the z-table
Noah -
on the z-table
Alex -
Noah -
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Find the area under the standard normal curve between Z=1.5 and Z=2.4.
Find the area under the standard normal curve between Z=1.5 and Z=2.4.
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Which of the following is a sampling distribution?
Which of the following is a sampling distribution?
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The correct answer is the distribution of average height statistics that could happen from all possible samples of college students. Remember that a sampling distribution isn't just a statistic you get form taking a sample, and isn't just a piece of data you get from doing sampling. Instead, a sampling distribution is a distribution of sample statistics you could get from all of the possible samples you might take from a given population.
The correct answer is the distribution of average height statistics that could happen from all possible samples of college students. Remember that a sampling distribution isn't just a statistic you get form taking a sample, and isn't just a piece of data you get from doing sampling. Instead, a sampling distribution is a distribution of sample statistics you could get from all of the possible samples you might take from a given population.
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Given a fair coin, what is the probability of obtaining 5 heads and 3 tails from 8 tosses?
Given a fair coin, what is the probability of obtaining 5 heads and 3 tails from 8 tosses?
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First, there are 8 trials and either choose 5 or 3 for heads or tails, respectively. Using this knowledge: 
. Next, the chance for either heads or tails is 0.5 and there are 5 heads and 3 tails. Thus: 
. Multiply: 
and 
and obtain 0.2188.
First, there are 8 trials and either choose 5 or 3 for heads or tails, respectively. Using this knowledge:
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There are 52 total cards in a full deck of playing cards. If a card dealer chooses 4 cards from the deck at random and without replacement, what is the chance that the dealer draws four kings as the first four cards?
There are 52 total cards in a full deck of playing cards. If a card dealer chooses 4 cards from the deck at random and without replacement, what is the chance that the dealer draws four kings as the first four cards?
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In a normal deck of playing cards, there are 4 kings. Thus, when the dealer draws the first card, the chance of the dealer obtaining a king is 4 out of 52. Because this card has been picked and is not replaced, the chance that the next card chosen is a king is 3 out of 52. The chance the third card is a king is 2 out of 52 and the fourth card is 1 out of 52. Each of these events is multiplied together, thus obtaining the correct answer, 0.0000037.
In a normal deck of playing cards, there are 4 kings. Thus, when the dealer draws the first card, the chance of the dealer obtaining a king is 4 out of 52. Because this card has been picked and is not replaced, the chance that the next card chosen is a king is 3 out of 52. The chance the third card is a king is 2 out of 52 and the fourth card is 1 out of 52. Each of these events is multiplied together, thus obtaining the correct answer, 0.0000037.
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Let us suppose we have population data where the data are distributed Poisson
(see the figure for an example of a Poisson random variable). 
Which distribution increasingly approximates the sample mean as the sample size increases to infinity?
Let us suppose we have population data where the data are distributed Poisson
(see the figure for an example of a Poisson random variable).
Which distribution increasingly approximates the sample mean as the sample size increases to infinity?
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The Central Limit Theorem holds that for any distribution with finite mean and variance the sample mean will converge in distribution to the normal as sample size 
.
The Central Limit Theorem holds that for any distribution with finite mean and variance the sample mean will converge in distribution to the normal as sample size
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A survey company samples 60 randomly selected college students to see if they own an American Express credit card. One percent of all college students own an American Express credit card. Does the Central Limit Theorem apply?
A survey company samples 60 randomly selected college students to see if they own an American Express credit card. One percent of all college students own an American Express credit card. Does the Central Limit Theorem apply?
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No. Whenever we get a "proportion" question we need to check whether 
and whether 
.
In this problem, 
.
Therefore,
.
So the central limit theorem does not apply.
No. Whenever we get a "proportion" question we need to check whether
In this problem,
Therefore,
So the central limit theorem does not apply.
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A fair coin is flipped three times and comes up heads each time. What is the probability that the fourth toss will also come up heads?
A fair coin is flipped three times and comes up heads each time. What is the probability that the fourth toss will also come up heads?
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Remember, no matter what the previous trials' results, the probability of a head (or tail) does not change from 
because each trial is independent of the others.
Remember, no matter what the previous trials' results, the probability of a head (or tail) does not change from
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Which parameters define the normal distribution?
Which parameters define the normal distribution?
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The two main parameters of the normal distribution are 
and 
. 
is a location parameter which determines the location of the peak of the normal distribution on the real number line. 
is a scale parameter which determines the concentration of the density around the mean. Larger 
's lead the normal to spread out more than smaller 
's.
The two main parameters of the normal distribution are
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