How does the decrease in confidence affect the sample size required?

All AP Statistics Resources

Which of the following statements are correct about confidence intervals? 

Possible Answers:

The width of a confidence interval increases as the sample size increases and increases as the confidence level increases.

The width of a confidence interval decreases as the sample size increases and increases as the confidence level decreases.

The width of a confidence interval decreases as the sample size increases and increases as the confidence level increases.

The width of a confidence interval does not change as the sample size increases and increases as the confidence level increases.

The width of a confidence interval increases as the sample size increases and increases as the confidence level decreases.

Correct answer:

The width of a confidence interval decreases as the sample size increases and increases as the confidence level increases.

Explanation:

Larger samples give narrower intervals. We are able to estimate a population proportion more precisely with a larger sample size. 

As the confidence level increases the width of the confidence interval also increases. A larger confidence level increases the chance that the correct value will be found in the confidence interval. This means that the interval is larger. 

You are asked to create a  confidence interval with a margin of error no larger than

Correct answer:

Explanation:

Keep in mind that the margin of error for a confidence interval based on a normal population is equal to , where  is the -score corresponding to the desired confidence level.

From the problem, we can tell that  and . We can then solve for  algebraically:

The minimum sample size is  rounded up, which is . If you are unsure on problems like these, you can check the margin of error for your answer rounded down and then rounded up (in this case, for  and .)

Jim calculated a  confidence interval for the mean height in inches of boys in his high school. He is not sure how to interpret this interval. Which of the following explains the meaning of  confidence.

Possible Answers:

There is a  chance that Jim's interval contains the true mean height.

In the long run,  of all confidence intervals calculated from the same population will contain the true mean height.

 of boys' heights fall with the interval Jim calculated.

 of all possible sample means fall within Jim's interval.

More information is needed.

Correct answer:

In the long run,  of all confidence intervals calculated from the same population will contain the true mean height.

Explanation:

95% confidence means that the methods Jim uses to calculate his confidence interval give him correct results 95% of the time. It does not mean that there is a 95% chance that the true mean will be inside the interval. It also does not mean that 95% of all heights or possible sample means fall within the interval.

The number of hamburgers served by McGregors per day is normally distributed and has a mean of  hamburgers and a standard deviation of  Find the range of customers served on the middle  percent of days. 

Correct answer:

Explanation:

First, find the first quartile of the distribution.

Then, find the third quartile of the distribution. 

The probability that it will rain today is 0.35. What is the probability that it will not rain?

Correct answer:

Explanation:

The answer is 0.65 because Pr(~Rain) is the complement of Pr(Rain) and both events are mutually exclusive.

When two events are mutually exclusive, . Since probabilities must sum up to 1, this implies that . 

Assume there is an election involving three parties: D, R, and I. The probability of D winning is .11, R winning is .78, and I winning is .11. What is the probability of D or R winning? 

Explanation:

Since all of the events are mutually exclusive (one of the parties must win), you can get the probability of either D or R winning by adding their probabilities. 

Since the probability of D winning is .11 and R winning is .78, the probability of D or R winning is .89. 

If 1 card is chosen at random from a deck of cards, what is the probability that it will be a heart or a king?

Possible Answers:

29/52

4/13

17/52

26/52

13/52

Explanation:

In a deck of cards, there are 52 total cards, 13 hearts, 4 kings, and 1 king that is a heart.

So, 

Using a standard deck of cards, what is the probability of choosing a single face card?

Correct answer:

A student randomly selected a highlighter from her desk.  There were five highlighters on the desk, each of a different color--blue, green, yellow, red, and orange.  What is the probability that the student selected either the red or the yellow highlighter?

Correct answer:

Explanation:

In this case, we want to know the probability of multiple, mutually exclusive possible outcomes. To determine the probability of the two possible outcomes, simply add them together.  This is called the addition rule.

What is the probability of obtaining at most  heads when tossing a fair coin  times?

Correct answer:

Explanation:

First find the number of ways to get 0, 1, or 2 heads.

Remember, .Also, and .

 heads:

 head:

 heads:

Now combine these to find the probability of seeing at most 2 heads in 10 coin tosses:

All AP Statistics Resources

How does confidence level affect sample size?

What happens to the sample size as the confidence interval decreases?

What happens when confidence level decreases?

Does increased sample size increase confidence?