Significance - AP Statistics
Card 1 of 168
You and a classmate wanted to test the effect of sugars and fats on levels of blood sugar.
Your classmate told you that they found the null hypothesis valid, which was what there is no difference between the effects of sugars and fats on blood sugar levels.
If the null hypothesis was actually false, what type of error was made?
You and a classmate wanted to test the effect of sugars and fats on levels of blood sugar.
Your classmate told you that they found the null hypothesis valid, which was what there is no difference between the effects of sugars and fats on blood sugar levels.
If the null hypothesis was actually false, what type of error was made?
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A type I error occurs when the null hypothesis is valid but rejected.
A type II error occurs when the null hypothesis is false, but fails to be rejected.
Because the null hypothesis was false, but had failed to be rejected, they made a Type II error.
A type I error occurs when the null hypothesis is valid but rejected.
A type II error occurs when the null hypothesis is false, but fails to be rejected.
Because the null hypothesis was false, but had failed to be rejected, they made a Type II error.
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You and a friend wanted to test the effect of similar servings of juice and soda on blood sugar levels.
Your friend told you that they found the null hypothesis valid, which was what there is no difference between the effects of similar servings of juice and soda on blood sugar levels.
If the null hypothesis was actually false, what type of error was made?
You and a friend wanted to test the effect of similar servings of juice and soda on blood sugar levels.
Your friend told you that they found the null hypothesis valid, which was what there is no difference between the effects of similar servings of juice and soda on blood sugar levels.
If the null hypothesis was actually false, what type of error was made?
Tap to reveal answer
A type I error occurs when the null hypothesis is valid but rejected.
A type II error occurs when the null hypothesis is false, but fails to be rejected.
Because the null hypothesis was false, but had failed to be rejected, they made a Type II error.
A type I error occurs when the null hypothesis is valid but rejected.
A type II error occurs when the null hypothesis is false, but fails to be rejected.
Because the null hypothesis was false, but had failed to be rejected, they made a Type II error.
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A factory claims that only 1% of their widgets are defective but a large amount of their produced widgets have been breaking for customers. A test is conducted to figure out if the factory claim of 1% defective is true or if the customers claim of graeater than 1% is true. What would be an example of a Type II error?
A factory claims that only 1% of their widgets are defective but a large amount of their produced widgets have been breaking for customers. A test is conducted to figure out if the factory claim of 1% defective is true or if the customers claim of graeater than 1% is true. What would be an example of a Type II error?
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Type II Error is not rejecting a truly false null hypothesis. This means that the test supports the factory claim of 1% even though the true amount is more than that.
Type II Error is not rejecting a truly false null hypothesis. This means that the test supports the factory claim of 1% even though the true amount is more than that.
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A pretzel company advertises that their pretzels contain less than 1.0g of of sodium per serving. You take a simple random sample of 10 pretzel servings, and calculate that the mean amount of sodium is 1.20 g, with a standard deviation of 0.1 g.
At the 95% confidence level, does your sample suggest that the pretzels actually have higher than 1.0g of sodium per serving?
A pretzel company advertises that their pretzels contain less than 1.0g of of sodium per serving. You take a simple random sample of 10 pretzel servings, and calculate that the mean amount of sodium is 1.20 g, with a standard deviation of 0.1 g.
At the 95% confidence level, does your sample suggest that the pretzels actually have higher than 1.0g of sodium per serving?
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This is a one- tailed t-test. It is one-tailed because the question asks whether the pretzel's mean is actually higher, so we are only interested in the right hand tail. We will be using the t-distribution because the population standard deviation is not known.
First we write our hypotheses:
Now we need the appropriate formula for a t-test. We will be using standard error because we are working with the standard deviation of a sampling distribution.
Now we fill in the values from our problem
Now we must look up the t-critical value, or use technology to find the p-value.
We must find the t-critcal value by finding 
Because our test statistic 6.32 is more extreme than our critical value, we reject our null hypothesis and conclude that the pretzels do have a higher mean than 1.0.
If you calculated a p-value using technology, p=0.00006884.
Because 
, we reject our null hypothesis and conclude that the pretzels do have a higher mean than 1.0 g.
This is a one- tailed t-test. It is one-tailed because the question asks whether the pretzel's mean is actually higher, so we are only interested in the right hand tail. We will be using the t-distribution because the population standard deviation is not known.
First we write our hypotheses:
Now we need the appropriate formula for a t-test. We will be using standard error because we are working with the standard deviation of a sampling distribution.
Now we fill in the values from our problem
Now we must look up the t-critical value, or use technology to find the p-value.
We must find the t-critcal value by finding
Because our test statistic 6.32 is more extreme than our critical value, we reject our null hypothesis and conclude that the pretzels do have a higher mean than 1.0.
If you calculated a p-value using technology, p=0.00006884.
Because
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James goes to UCLA, and he believes that the atheletes of UCLA are better runners, than the country average. He did a bit of a research and found that the national average time for a two-mile run for college atheletes is 
min with a standard deviation of 
minute. He then sampled 
UCLA atheletes and found that their average two-mile time was 
minutes.
Is James' data statistically significant? Can we confirm that UCLA atheletes are better than average runners? And if so, to which level of certainty: 
, 
, 
, 
James goes to UCLA, and he believes that the atheletes of UCLA are better runners, than the country average. He did a bit of a research and found that the national average time for a two-mile run for college atheletes is
Is James' data statistically significant? Can we confirm that UCLA atheletes are better than average runners? And if so, to which level of certainty:
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Using a Z-test (we have population SD, not sample SD) and a population of 
, we arrive at a P-value of 
, which is lower than 
, but above 
.
Using a Z-test (we have population SD, not sample SD) and a population of
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Bob and Alvin suspect that the weight of the average man in Jackson, Mississippi is significantly different than the weight of the average man in Green Bay, Wisconsin. They sample 100 men in Jackson and find that the average weight is 191 lbs, and the sample standard deviation is 30 lbs.
They sample 100 men in Green Bay and find that the average weight is 184, and the sample standard deviation is 25.
Bob says that they should present a 95% 1-tailed test where the alternate hypothesis is:
and the null hypothesis is:
Alvin disagrees; he says that a one-tailed test assumes that we already suspected a higher weight in Jackson. He recommends a 95% 2-tailed test where
and
.
Show why the 1-tailed test rejects its null hypothesis and the 2-tailed test fails to reject its null hypothesis. Provide the following:
-
The Z-value for a 1-tailed 95% test.
-
The Z-value for a 2-tailed 95% test.
-
The Z-value for the sample difference of 7.
Bob and Alvin suspect that the weight of the average man in Jackson, Mississippi is significantly different than the weight of the average man in Green Bay, Wisconsin. They sample 100 men in Jackson and find that the average weight is 191 lbs, and the sample standard deviation is 30 lbs.
They sample 100 men in Green Bay and find that the average weight is 184, and the sample standard deviation is 25.
Bob says that they should present a 95% 1-tailed test where the alternate hypothesis is:
and the null hypothesis is:
Alvin disagrees; he says that a one-tailed test assumes that we already suspected a higher weight in Jackson. He recommends a 95% 2-tailed test where
and
Show why the 1-tailed test rejects its null hypothesis and the 2-tailed test fails to reject its null hypothesis. Provide the following:
-
The Z-value for a 1-tailed 95% test.
-
The Z-value for a 2-tailed 95% test.
-
The Z-value for the sample difference of 7.
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The variance for the sample difference is:
The standard deviation for the sample difference is:
The Z-value for a difference of 7 is:
The sample difference Z-value is greater than the 1-tailed Z-value (causing us to reject the null hypothesis) and is less than the 2-tailed Z-value (causing us to fail to reject the null hypothesis).
The variance for the sample difference is:
The standard deviation for the sample difference is:
The Z-value for a difference of 7 is:
The sample difference Z-value is greater than the 1-tailed Z-value (causing us to reject the null hypothesis) and is less than the 2-tailed Z-value (causing us to fail to reject the null hypothesis).
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Jimmy thinks that Josh cannot shoot more than 50 points on average in a game. Josh disputes this claim and tells Jimmy that he is going to play 10 games and prove him wrong. What is the null hypothesis?
Jimmy thinks that Josh cannot shoot more than 50 points on average in a game. Josh disputes this claim and tells Jimmy that he is going to play 10 games and prove him wrong. What is the null hypothesis?
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The null hypothesis is what we intend to either reject or fail to reject using our sample data. In this case, the null hypothesis is that Josh cannot shoot more than 50 points on average, and Josh's performance in 10 games are the sample data we use to assess this hypothesis.
The null hypothesis is what we intend to either reject or fail to reject using our sample data. In this case, the null hypothesis is that Josh cannot shoot more than 50 points on average, and Josh's performance in 10 games are the sample data we use to assess this hypothesis.
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A student is beginning an analysis to determine whether there is a relationship between temperatures and traffic accidents. The student is trying to articulate a null hypothesis for the study. Which of the following is an acceptable null hypothesis?
A student is beginning an analysis to determine whether there is a relationship between temperatures and traffic accidents. The student is trying to articulate a null hypothesis for the study. Which of the following is an acceptable null hypothesis?
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The null hypothesis is the default hypothesis and predicts that there is no relationship between the variables in question. Each of the incorrect answer choices here either predicts a relationship between variables or makes a broad assertion that includes much more than the variables in question.
The null hypothesis is the default hypothesis and predicts that there is no relationship between the variables in question. Each of the incorrect answer choices here either predicts a relationship between variables or makes a broad assertion that includes much more than the variables in question.
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A statistician has determined that she will reject the null hypothesis if she can have 
confidence that there is a statistically significant relationship between the variables in question. She conducts a statistical analysis and obtains a p value of 
. Should the statistician reject the null hypothesis?
A statistician has determined that she will reject the null hypothesis if she can have
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The statistician has determined that she will only reject the null hypothesis if she has 95% confidence that there is a relationship between variables.
To have this level of confidence, the statistician must obtain a p value of 0.05 or lower.
Therefore, she should not reject the null hypothesis since 0.1 is greater that 0.05.
The statistician has determined that she will only reject the null hypothesis if she has 95% confidence that there is a relationship between variables.
To have this level of confidence, the statistician must obtain a p value of 0.05 or lower.
Therefore, she should not reject the null hypothesis since 0.1 is greater that 0.05.
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We are comparing the Democratic percentage of Detroit to the Democratic percentage of Dallas.
In Detroit, we sampled 300 people and 208 were Democrats (.693)
In Dallas, we sampled 350 people and 265 were Democrats (.757)
What is the p-value for the .064 difference?
(Assume a 2-tailed test.)
We are comparing the Democratic percentage of Detroit to the Democratic percentage of Dallas.
In Detroit, we sampled 300 people and 208 were Democrats (.693)
In Dallas, we sampled 350 people and 265 were Democrats (.757)
What is the p-value for the .064 difference?
(Assume a 2-tailed test.)
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Overall percentage = 
.0344 is for one tail, so .0688 is for both tails.
(Percentages are different at the .9312 level.)
Overall percentage =
.0344 is for one tail, so .0688 is for both tails.
(Percentages are different at the .9312 level.)
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No explanation available
No explanation available
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Suppose you conduct a paired 
-test to assess whether two group means significantly differ and find a 
-score of 1.645. At what alpha level would this cause you to reject the null?
Suppose you conduct a paired
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The critical value for a 
-test with alpha set to 0.10 is 1.645.
The critical value for a
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Bob wants to statistically determine if the mean height of middle school boys is greater than the mean height of middle school girls. He wants to use a significance level of 
What must be true for him to reject 
. 
is known.
Bob wants to statistically determine if the mean height of middle school boys is greater than the mean height of middle school girls. He wants to use a significance level of
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Step 1: We need to use a 2-sample z test because there are 2 samples, boys and girls. The population standard deviation, 
, is known, so we can assume a standard distribution for each sample.
Step 2: This is a one-sided z test because the questions asks if the mean height of boys is greater than the mean height of girls.
Step 3: significance level, or alpha, is 
. This means we need a p-value less than 
in order to reject the null hypothesis 
. A z-score greater than 
would ensure a p-value less than 
.
Step 1: We need to use a 2-sample z test because there are 2 samples, boys and girls. The population standard deviation,
Step 2: This is a one-sided z test because the questions asks if the mean height of boys is greater than the mean height of girls.
Step 3: significance level, or alpha, is
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No explanation available
No explanation available
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A statistician conducts a regression analysis and obtains a p-value of 0.1. It is more likely than not that there is a relationship between the variables in the study.
A statistician conducts a regression analysis and obtains a p-value of 0.1. It is more likely than not that there is a relationship between the variables in the study.
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A p-value of 0.1 is generally not sufficient to reject the null hypothesis, but this is only because we want a high degree of confidence before finding a relationship between variables. Here, there is most likely a relationship between the variables even though the statistician could not reject the null hypothesis.
A p-value of 0.1 is generally not sufficient to reject the null hypothesis, but this is only because we want a high degree of confidence before finding a relationship between variables. Here, there is most likely a relationship between the variables even though the statistician could not reject the null hypothesis.
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Which of the following is an incorrect condition requirement for regression inference?
Which of the following is an incorrect condition requirement for regression inference?
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All of the following choices are correct conditions except for the choice concerning a trend/pattern of some sort in the residual plot. For regression inference to be accurate, we need to look at the residual plot of the data of interest and make sure there is random scatter. Random scatter indicates that the ordered pairs are indeed independent of each other. Any sort of pattern present in the residual plot would not satisfy that requirement, and therefore would not enable us to successfully use regression inference.
All of the following choices are correct conditions except for the choice concerning a trend/pattern of some sort in the residual plot. For regression inference to be accurate, we need to look at the residual plot of the data of interest and make sure there is random scatter. Random scatter indicates that the ordered pairs are indeed independent of each other. Any sort of pattern present in the residual plot would not satisfy that requirement, and therefore would not enable us to successfully use regression inference.
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For a data set, the least-squares regression line has a 
confidence interval for the slope of 
.
Based on this confidence interval, what can you do with a hypothesis test at 
significance level where 
and 
?
For a data set, the least-squares regression line has a
Based on this confidence interval, what can you do with a hypothesis test at
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Notice that the interval does not include 
. This means that the P-value for the hypothesis test would be under 5%, which would lead us to reject our null hypothesis.
Any confidence interval can be used to create a hypothesis test by inverting it, and it is fairly simple, but the concept is tested into graduate-level statistics theory.
Notice that the interval does not include
Any confidence interval can be used to create a hypothesis test by inverting it, and it is fairly simple, but the concept is tested into graduate-level statistics theory.
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A prominent football coach is being reviewed for his performance in the past season. To evaluate how well the coach has done, the team manager runs a statistical test comparing the coach to a sample of coaches in the league. If the test suggests that the coach outperformed other coaches when in fact he did not, and the manager then rejects the null hypothesis (that the coach did not outperform the other coaches), what kind of error is he committing?
A prominent football coach is being reviewed for his performance in the past season. To evaluate how well the coach has done, the team manager runs a statistical test comparing the coach to a sample of coaches in the league. If the test suggests that the coach outperformed other coaches when in fact he did not, and the manager then rejects the null hypothesis (that the coach did not outperform the other coaches), what kind of error is he committing?
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A type I error occurs when one rejects a null hypothesis that is in fact true. The null hypothesis is that the coach does not outperform other coaches, and the test reccomends that we reject it even though it is true. Thus, a type I error has been committed.
A type I error occurs when one rejects a null hypothesis that is in fact true. The null hypothesis is that the coach does not outperform other coaches, and the test reccomends that we reject it even though it is true. Thus, a type I error has been committed.
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If a hypothesis test uses a 
confidence level, then what is its probability of Type I Error?
If a hypothesis test uses a
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By definition, the probability of Type I Error is,
where,
represents Probability of Type I Error and 
represents the confidence level.
Thus resulting in:
By definition, the probability of Type I Error is,
where,
Thus resulting in:
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For significance tests, which of the following is an incorrect way to increase power (the probability of correctly rejecting the null hypothesis)?
For significance tests, which of the following is an incorrect way to increase power (the probability of correctly rejecting the null hypothesis)?
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Recall that power is 
. The probability of Type I and Type II errors will change inversely of each other as the probability of making a Type I error changes. If 
increases, then 
decreases, and as a result power will increase. So if 
decreases, 
would increase, and power would decrease; therefore decreasing 
will not increase power.
Recall that power is
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