### All AP Statistics Resources

## Example Questions

### Example Question #11 : Inference

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?

**Possible Answers:**

Type I and II

Type II

Type I

Neither

**Correct answer:**

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 #11 : Inference

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?

**Possible Answers:**

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 shows that the percentage of defective widgets is 1% and the factory claim is upheld.

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.

**Correct answer:**

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 : How To Define A Type I Error

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?

**Possible Answers:**

None of the above

Type II error

Type M error

Type I error

**Correct answer:**

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 : How To Define A Type I Error

If a hypothesis test uses a confidence level, then what is its probability of Type I Error?

**Possible Answers:**

**Correct answer:**

By definition, the probability of Type I Error is,

where,

represents Probability of Type I Error and represents the confidence level.

Thus resulting in:

### Example Question #1 : How To Define A Type I Error

For significance tests, which of the following is an incorrect way to increase power (the probability of correctly rejecting the null hypothesis)?

**Possible Answers:**

Decrease the probability of making a Type I error

Decrease variability of samples

Increase sample size

Increase the difference between the null hypothesis and the alternative hypothesis

Increase the probability of making a Type I error

**Correct answer:**

Decrease the probability of making a Type I error

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.

### Example Question #1 : How To Define A Type I Error

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?

**Possible Answers:**

Type II

Neither

Type I and II

Type I

**Correct answer:**

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?

**Possible Answers:**

Type I and II

Type II

Type I

Neither

**Correct answer:**

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?

**Possible Answers:**

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.

There are 12 ounces in each bag and the test backs the company claim.

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.

**Correct answer:**

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.

### Example Question #1 : How To Find P Values

and the sample mean is 12.

Select the answer so that both statements indicate a rejection of the null hypothesis at the 95% confidence level.

**Possible Answers:**

1) The Z-value for 12 is less than 1.96

and

2) The p-value for 12 is greater than .05

1) The Z-value for 12 is greater than 1.96

and

2) The p-value for 12 is less than .05

1) The Z-value for 12 is less than 1.96

and

2) The p-value for 12 is less than .05

1) The Z-value for 12 is greater than 1.96

and

2) The p-value for 12 is greater than .05

**Correct answer:**

1) The Z-value for 12 is greater than 1.96

and

2) The p-value for 12 is less than .05

In order to reject the null hypothesis, the Z-value for the sample must be greater than (i.e. must lie outside of) the Z-value of the confidence level.

By definition, if the Z-value of the sample is greater than the Z-value of the confidence level, then the p-value of the sample must be less than the p-value for the confidence level.

### Example Question #1 : How To Find P Values

Under the null hypothesis, the distribution of a stock price is normal with mean and standard deviation . The actual stock price now is . What is the probability that the stock price is this much or greater under the null hypothesis?

**Possible Answers:**

**Correct answer:**

This exercise consists of computing the p-value. The null distribution is normal, so we must compute the z-score with the actual data we have and use it to compute the p-value.

We have:

Now we calculate the chance that , using the context from the problem.

Using a normal table, we get , which is the answer.