# AP Statistics : Inference

## Example Questions

### 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?

Explanation:

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.

### Example Question #41 : Ap Statistics

A study would like to determine whether meditation helps students improve focus time. They used a control group of 30 students and compared their focus time to a group of 30 meditating students and compared their average time spent meditating. What is the appropriate null hypothesis for this study?

Explanation:

Because we are comparing two samples, the hypothesis takes the form of Mu1- Mu2. Because we are testing the claim that meditation increases average study time, the null hypothesis must cover all other outcomes. That means the null hypothesis is that the difference between control minus meditation is greater than or equal to zero.

### Example Question #1 : Estimation

What is , or the expected value of  for any distribution?

Explanation:

is , or the mean, of the population. This makes sense since  literally means the expected value of . The mean is the expected value of

Explanation:

### Example Question #1 : How To Estimate Margins Of Errors

The next local election is predicted to have a  split in votes for the top two candidates. How many people should be polled to obtain a margin of error of 3% at the 95% confidence level?

Explanation:

If you remember the required formula, this problem is rather simple. Plug in the given numbers and simplify:

If you don't remember the formula, this problem is more challenging. Your best bet in this case is to construct the confidence interval and rearrange to solve for the required sample size.

### Example Question #2 : How To Estimate Margins Of Errors

When dealing with confidence intervals, the margin of error gets smaller when z* gets ________ and n gets _________.

smaller . . . larger

smaller . . . smaller

larger . . . smaller

There is no way to affect the margin of error in confidence intervals

larger . . . larger

smaller . . . larger

Explanation:

For confidence intervals, a small margin of error is preferred, as it indicates that the parameter of interest has been narrowed down to a precise interval. Having a large sample population to work from as well as a small z* (z* gets smaller as the confidence level percent gets lower) can help obtain a small margin of error.

Explanation:

Explanation:

### Example Question #1 : How To Find Confidence Intervals For A Mean

Suppose you have a normally distributed variable with known variance. How many standard errors do you need to add and subtract from the sample mean so that you obtain 95% confidence intervals?

Explanation:

To obtain 95% confidence intervals for a normal distribution with known variance, you take the mean and add/subtract . This is because 95% of the values drawn from a normally distributed sampling distribution lie within 1.96 standard errors from the sample mean.

### Example Question #1 : Estimation

An automotive engineer wants to estimate the cost of repairing a car that experiences a 25 MPH head-on collision. He crashes 24 cars, and the average repair is $11,000. The standard deviation of the 24-car sample is$2,500.

Provide a 98% confidence interval for the true mean cost of repair.