Statistical Patterns and Random Phenomena - AP Statistics
Card 1 of 560
Suppose that the mean height of college students is 70 inches with a standard deviation of 5 inches. If a random sample of 60 college students is taken, what is the probability that the sample average height for this sample will be more than 71 inches?
Suppose that the mean height of college students is 70 inches with a standard deviation of 5 inches. If a random sample of 60 college students is taken, what is the probability that the sample average height for this sample will be more than 71 inches?
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First check to see if the Central Limit Theorem applies. Since n > 30, it does. Next we need to calculate the standard error. To do that we divide the population standard deviation by the square-root of n, which gives us a standard error of 0.646. Next, we calculate a z-score using our z-score formula:
Plugging in gives us:
Finally, we look up our z-score in our z-score table to get a p-value.
The table gives us a p-value of,
First check to see if the Central Limit Theorem applies. Since n > 30, it does. Next we need to calculate the standard error. To do that we divide the population standard deviation by the square-root of n, which gives us a standard error of 0.646. Next, we calculate a z-score using our z-score formula:
Plugging in gives us:
Finally, we look up our z-score in our z-score table to get a p-value.
The table gives us a p-value of,
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A researcher wants to determine whether there is a significant linear relationship between time spent meditating and time spent studying. What is the appropriate null hypothesis for this study?
A researcher wants to determine whether there is a significant linear relationship between time spent meditating and time spent studying. What is the appropriate null hypothesis for this study?
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This question is about a linear regression between time spent meditating and time spent studying. Therefore, the hypothesis is regarding Beta1, the slope of the line. We are testing a non-directional or bi-directional claim that the relationship is significant. Therefore, the null hypothesis is that the relationship is not significant, meaning the slope of the line is equal to zero.
This question is about a linear regression between time spent meditating and time spent studying. Therefore, the hypothesis is regarding Beta1, the slope of the line. We are testing a non-directional or bi-directional claim that the relationship is significant. Therefore, the null hypothesis is that the relationship is not significant, meaning the slope of the line is equal to zero.
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The president of a country is trying to estimate the average income of his citizens. He randomly samples residents and collects information about their salaries. A 
percent confidence interval computed from this data for the mean income per citizen is 
Which of the following provides the best interpretation of this confidence interval?
The president of a country is trying to estimate the average income of his citizens. He randomly samples residents and collects information about their salaries. A
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A confidence interval is a statement about the mean of the population the sample is drawn from so there is a 
percent probability that a 
percent confidence interval contains the true mean of the population.
A confidence interval is a statement about the mean of the population the sample is drawn from so there is a
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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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In a particular library, there is a sign in the elevator that indicates a limit of 
persons and a weight limit of 
. Assume an approximately normal distribution, that the average weight of students, faculty, and staff on campus is 
, and the standard deviation is 
.
If a random sample of 
people is taken, what is the standard deviation of their weights?
In a particular library, there is a sign in the elevator that indicates a limit of 
If a random sample of 
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This question deals with the Central Limit Theorem, which states that a random sample taken from a large population where the sampling distribution of sample averages is approximately normal has a standard deviation equal to the standard deviation of the population divided by the square root of the sample size. The information given allows us to apply the Central Limit Theorem as it satisfies the necessary characteristics of the sampling distribution/size. The standard deviation of the population is 27lbs, and the sample size is 36; therefore, the standard deviation of the 36-person random sample is 
, which gives us 4.5lbs.
This question deals with the Central Limit Theorem, which states that a random sample taken from a large population where the sampling distribution of sample averages is approximately normal has a standard deviation equal to the standard deviation of the population divided by the square root of the sample size. The information given allows us to apply the Central Limit Theorem as it satisfies the necessary characteristics of the sampling distribution/size. The standard deviation of the population is 27lbs, and the sample size is 36; therefore, the standard deviation of the 36-person random sample is
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Cat owners spend an average of 
per month on their pets, with a standard deviation of 
.
What is the probability that a randomly chosen cat owner spends more than 
a month on their pet?
Cat owners spend an average of
What is the probability that a randomly chosen cat owner spends more than
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First, draw the appropriate curve to represent the problem:
Because we want greater than 45, shade the right hand portion of the graph.
We will be using the z-distribution and not the t-distribution because we know the population standard deviation.
We will find the z-score for this cat owner, and then find the area to the right of that z-score to find the probability of spending more than $45.
We begin with the z-score formula:
Now find each value from the information given:
Fill in the formula
Now look this z-score up in the z-table to find the area under the curve. Look up 1.0 in the row and 0.00 in the column.
If your z table gives area shaded to the left, you will find
You want the area to the right, so subtract the above from 1.
If your z table gives area shaded to the right, you will find
Because you want the area to the right of z=1.0, this is your answer.
First, draw the appropriate curve to represent the problem:
Because we want greater than 45, shade the right hand portion of the graph.
We will be using the z-distribution and not the t-distribution because we know the population standard deviation.
We will find the z-score for this cat owner, and then find the area to the right of that z-score to find the probability of spending more than $45.
We begin with the z-score formula:
Now find each value from the information given:
Fill in the formula
Now look this z-score up in the z-table to find the area under the curve. Look up 1.0 in the row and 0.00 in the column.
If your z table gives area shaded to the left, you will find
You want the area to the right, so subtract the above from 1.
If your z table gives area shaded to the right, you will find
Because you want the area to the right of z=1.0, this is your answer.
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Cat owners spend an average of $40 per month on their pets, with a standard deviation of $5.
What is the probability that a randomly selected pet owner spends less than 
a month on their pet?
Cat owners spend an average of $40 per month on their pets, with a standard deviation of $5.
What is the probability that a randomly selected pet owner spends less than
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First, draw the distribution and the area you are interested in.
Next, calculate the z-score for the person of interest. Because the population standard deviation is known, we will use the formula for z-score and not t-score.
We will find the z-score for the person of interest, and then calculate the area under the curve that falls below, or to the left of that z-score.
Now we will find the value for each variable given in the problem:
Third, solve for z using the information in the problem.
Now we must determine the area under the curve to the left of a z-score of 1.0. We will consult a z-table.
Look up 1.0 in the row and 0.00 in the column.
If your z-table gives shaded area to the left, you will get
We are interested in area to the left, which is what we found, so this is our answer.
If your z-table gives shaded area to the right, you will get
Because we want the area to the left of z=1.0, we will subtract that area from 1:
First, draw the distribution and the area you are interested in.
Next, calculate the z-score for the person of interest. Because the population standard deviation is known, we will use the formula for z-score and not t-score.
We will find the z-score for the person of interest, and then calculate the area under the curve that falls below, or to the left of that z-score.
Now we will find the value for each variable given in the problem:
Third, solve for z using the information in the problem.
Now we must determine the area under the curve to the left of a z-score of 1.0. We will consult a z-table.
Look up 1.0 in the row and 0.00 in the column.
If your z-table gives shaded area to the left, you will get
We are interested in area to the left, which is what we found, so this is our answer.
If your z-table gives shaded area to the right, you will get
Because we want the area to the left of z=1.0, we will subtract that area from 1:
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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 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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In order to be considered a normal distribution, a data set (when graphed) must follow a bell-shaped symmetrical curve centered around the mean.
It must also adhere to the empirical rule that indicates the percentage of the data set that falls within (plus or minus) 1, 2 and 3 standard deviations of the mean.
In order to be a normal distribution, what percentage of the data set must fall within:
-
-
-
In order to be considered a normal distribution, a data set (when graphed) must follow a bell-shaped symmetrical curve centered around the mean.
It must also adhere to the empirical rule that indicates the percentage of the data set that falls within (plus or minus) 1, 2 and 3 standard deviations of the mean.
In order to be a normal distribution, what percentage of the data set must fall within:
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- Percentile for Z=1 is .8413 - or - .1587 in one tail - or - .3174 in both tails -
1 - .3174=.6826
- Percentile for Z=2 is .9772 - or - .0228 in one tail - or - .0456 in both tails -
1 - .0456=.9544
- Percentile for Z=3 is .9987 - or - .0013 in one tail - or - .0026 in both tails -
1 - .0026=.9974
- Percentile for Z=1 is .8413 - or - .1587 in one tail - or - .3174 in both tails -
1 - .3174=.6826
- Percentile for Z=2 is .9772 - or - .0228 in one tail - or - .0456 in both tails -
1 - .0456=.9544
- Percentile for Z=3 is .9987 - or - .0013 in one tail - or - .0026 in both tails -
1 - .0026=.9974
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All normal distributions can be described by two parameters: the mean and the variance. Which parameter determines the location of the distribution on the real number line?
All normal distributions can be described by two parameters: the mean and the variance. Which parameter determines the location of the distribution on the real number line?
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The mean determines where the normal distribution lies on the real number line, while the variance determines the spread of the distribution.
The mean determines where the normal distribution lies on the real number line, while the variance determines the spread of the distribution.
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Consider a normal distribution with a mean of 
and a standard deviation of 
. Which of the following statements are true according to the Empirical Rule?
of observations are at least 
.
of observations are between 
and 
.
of observations are between 
and 
.
Consider a normal distribution with a mean of
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and 3) are true by definition of the Empirical Rule - also known as the 68-95-99.7 Rule. Using our information with mean of 100 and a standard deviation of 5 we can create a bell curve with 100 in the middle. One standard deviation out from the mean would give us a range from 95 to 105 and would be in our 68% section. If we go two standard deviations out from 100 we would get the range 90 to 110 thus lying in the 95% section. Lastly, when we go out 3 standard deviations we get a range of 85 to 115 thus falling within the 99.7% section.
-
can be deduced as true because it means that 68% of observations are between 95 and 105, and removing one of those bounds (namely, the upper one) adds the 16%, since 
, of observations larger than 105 leading to 68 + 16 = 84% of observations greater than 95.
-
and 3) are true by definition of the Empirical Rule - also known as the 68-95-99.7 Rule. Using our information with mean of 100 and a standard deviation of 5 we can create a bell curve with 100 in the middle. One standard deviation out from the mean would give us a range from 95 to 105 and would be in our 68% section. If we go two standard deviations out from 100 we would get the range 90 to 110 thus lying in the 95% section. Lastly, when we go out 3 standard deviations we get a range of 85 to 115 thus falling within the 99.7% section.
-
can be deduced as true because it means that 68% of observations are between 95 and 105, and removing one of those bounds (namely, the upper one) adds the 16%, since
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What is the relationship between the mean and the median in a normally distributed population?
What is the relationship between the mean and the median in a normally distributed population?
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A normal distrubtion is completely symmetrical and has not outliers. This means that the mean is not pulled to either side by outliers and should lie directly in the middle along with the median (the middle number of the distribution).
A normal distrubtion is completely symmetrical and has not outliers. This means that the mean is not pulled to either side by outliers and should lie directly in the middle along with the median (the middle number of the distribution).
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Which is the following is NOT a property of the normal distribution?
Which is the following is NOT a property of the normal distribution?
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The empirical rule tells us that the probability that a random data point is within one standard deviation of the mean is approximately 68%, not 78%.
The empirical rule tells us that the probability that a random data point is within one standard deviation of the mean is approximately 68%, not 78%.
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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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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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