### All AP Statistics Resources

## Example Questions

### Example Question #11 : Significance Logic And Establishing Hypotheses

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?

**Possible Answers:**

Josh cannot shoot more than 50 points.

Josh cannot shoot less than 50 points.

Josh cannot shoot exactly 50 points.

Josh cannot play 10 games.

**Correct answer:**

Josh cannot shoot more than 50 points.

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.

### Example Question #1 : How To Establish A 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?

**Possible Answers:**

There is a negative relationship between temperatures and traffic accidents

No variable can accurately predict whether traffic accidents will increase

There is no relationship between temperatures and frequency of traffic accidents

There is a positive relationship between temperatures and traffic accidents

Traffic accidents increase as temperatures decrease

**Correct answer:**

There is no relationship between temperatures and frequency of traffic accidents

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.

### Example Question #1 : How To Establish A Null Hypothesis

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?

**Possible Answers:**

No

Not enough information to make a decision.

Conditionally

Maybe

Yes

**Correct answer:**

No

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.

### Example Question #2 : How To Establish A Null Hypothesis

The Environmental Protection Agency (EPA) wants to test the pollution level of the Colorado River. If the pollution level is too high, the water will be stopped from going into drinking water pipelines. The EPA randomly chooses different spots along the river to collect water samples from, and then tests the samples for their pollution levels. Which of the following decisions would result from the type I error?

**Possible Answers:**

Closing the drinking water pipelines because of the endangered frog population.

Closing the drinking water pipelines when the pollution levels are higher than the allowed limit.

Keeping the drinking water pipelines open when the pollution levels are within the allowed limit.

Closing the drinking water pipelines for the river when the pollution levels are within the allowed limit.

Keeping the drinking water pipelines open when the pollution levels are higher than the allowed limit.

**Correct answer:**

Closing the drinking water pipelines for the river when the pollution levels are within the allowed limit.

The hypotheses tested here are:

The true mean pollution level is within the allowed limit.

The true mean pollution level is higher than the allowed limit.

The type I error occurs when the null hypothesis is rejected even though it is actually true. In this case, the type I error would be deciding that the mean pollution levels are higher than the allowed limit and closing the drinking water pipelines.

### Example Question #3 : How To Establish A Null Hypothesis

A study would like to determine whether meditation helps students improve focus time. They know that the average focus time of an American 4^{th} grader is 23 minutes. They then recruit 50 meditators and calculate their average focus time. What is the appropriate null hypothesis for this study?

**Possible Answers:**

**Correct answer:**

Because we are comparing one sample to a known mean, the hypothesis is compared to the known average, 23. We would like to test the claim that meditating will increase focus time, therefore our null hypothesis must include all other outcomes. Therefore, the null is that Mu is less than or equal to 23.

### Example Question #4 : How To Establish A Null Hypothesis

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?

**Possible Answers:**

**Correct answer:**

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.

### Example Question #1 : How To Establish A Null Hypothesis

A researcher wants to compare 3 different treatments to see if any of the treatments affects study time. The three treatments studied are control group, a group given vitamins, and a group given a placebo. They found that the average time spent studying with control students was 2 hours, with students given vitamins it was 3 hours, and with placebos students studied 5 hours. Which of the following is the correct null hypothesis?

**Possible Answers:**

**Correct answer:**

Because we are comparing more than 2 groups, we must use an ANOVA for this problem. For an ANOVA problem, the null hypothesis is that all of the groups’ means are the same.

### Example Question #1 : How To Establish A Null Hypothesis

A researcher wants to investigate the claim that taking vitamins will help a student study longer. First, the researcher collects 32 students who do not take vitamins and determines their time spent studying. Then, the 32 students are given a vitamin for 1 week. After 1 week of taking vitamins, students are again tested to determine their time spent studying. Which of the following is the correct null hypothesis?

**Possible Answers:**

**Correct answer:**

Because the same students are tested twice, this is a paired study, therefore we must use a hypothesis appropriate for a paired t-test. The hypothesis for a paired t-test regards the average of the differences between before and after treatment, called MuD. We are testing the claim that vitamins increase study time, which would mean that study time for vitamin users would be greater than that of the control. Therefore the null must include all other outcomes. The null hypothesis should state that the difference between before and after treatment is greater than or equal to zero.

### Example Question #1 : How To Establish A Null Hypothesis

For her school science project, Susy wants to determine whether the ants in her neighborhood have smaller colonies than average. Research tells her that the average Harvester colony has around 4,000 ants. She counts the number of ants in 5 colonies in her neighborhood and determines the average colony size to be 3,700 ants. What is the appropriate null hypothesis for her science project?

**Possible Answers:**

**Correct answer:**

Susy wants to know whether ants in her neighborhood have smaller colonies, so that will be her alternative hypothesis. Therefore her null hypothesis needs to cover all other outcomes, that the colony sizes are greater than or equal to average colony size of 4000 ants.

### Example Question #1 : How To Establish A Null Hypothesis

For his school science project, Timmy wants to determine whether the ants in his neighborhood have colonies that are sized differently than normal. His research shows that the average Harvester colony has around 4000 ants. He counts the number of ants in 5 colonies and determines that the average colony size is 3,700 ants. What is the appropriate null hypothesis for his science project?

**Possible Answers:**

**Correct answer:**

Timmy does not have a directional hypothesis, he only wants to know whether local ant colonies are different from average. Therefore he thinks the colonies could be bigger or smaller than average. This means his alternative hypothesis is that the ant colonies are NOT equal to the average colony size of 4000 ants. His null hypothesis must include all other outcomes, which in this case is that local ant colonies are equal to the average size of 4000 ants.

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