AP Statistics : Defining Errors
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Example Questions
Example Question #1 : How To Define A Type Ii Error
If a test has a power of 
From the statistical definition of power (of a test), the power is equal to 
Therefore our equation to solve becomes:
Example Question #1 : Significance
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?
Type II
Type I and II
Type I
Neither
Type II
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.
Example Question #1 : How To Define A Type Ii Error
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?
Type I
Type I and II
Type II
Neither
Type II
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.
Example Question #1 : How To Define 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?
The test shows that only 1% are defective when the truth is that more than 1% are defective. The null is upheld when it should be rejected.
The test shows that there are more than 1% defective even though the null of just 1% is actually true.
The test came up with no definitive answer.
More than 1% are shown to be defective and the reject the factory claim of only 1% defective.
The test shows that the percentage of defective widgets is 1% and the factory claim is upheld.
The test shows that only 1% are defective when the truth is that more than 1% are defective. The null is upheld when it should be rejected.
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.
Example Question #1 : Defining Errors
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?
Type II error
Type M error
None of the above
Type I error
Type I error
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.
Example Question #1 : Defining Errors
If a hypothesis test uses a 
By definition, the probability of Type I Error is,
where,
Thus resulting in:
Example Question #1 : Defining Errors
For significance tests, which of the following is an incorrect way to increase power (the probability of correctly rejecting the null hypothesis)?
Decrease the probability of making a Type I error
Increase the probability of making a Type I error
Increase the difference between the null hypothesis and the alternative hypothesis
Increase sample size
Decrease variability of samples
Decrease the probability of making a Type I error
Recall that power is 
Example Question #3 : Defining Errors
In a recent athletic study, your lab partner told you that they rejected the null hypothesis that the fabric of running shoes has no effect on the wearer's running times.
If the null hypothesis was actually valid, what type of error was made?
Neither
Type I and II
Type I
Type II
Type I
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 true, but rejected, they made a Type I error.
Example Question #1 : How To Define A Type I Error
In a recent academic study, your lab partner told you that they rejected the null hypothesis that the ionization of water has no effect on the rate of grass growth.
If the null hypothesis was actually valid, what type of error was made?
Neither
Type I
Type II
Type I and II
Type I
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 true, but rejected, they made a Type I error.
Example Question #1 : How To Define A Type I Error
A company claims that they have 12 ounces of potato chips in each of their bags of chips. A customer complaint is filed that they do not truly contain 12 ounces but actually contain less. A sampling test is conducted to see if the comapny measure is true or not. What would be an example of a Type I error?
There are 12 ounces in each bag and the test backs the company claim.
There test does not come up with a defined answer.
The test finds that there are 12 ounces in each but the true amount is less than that.
The test finds there are less than 12 ounces but there are actually 12 ounces.
The test finds that there is less than 12 ounces and the customer claim was indeed true.
The test finds there are less than 12 ounces but there are actually 12 ounces.
The type I error is rejecting the null hypothesis when it is actually true. The null here is 12 ounces per bag so a type I error would be rejecting the company claim even though there are 12 ounces per bag.
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