What happens to the confidence interval when the level of confidence increases and decreases?

Example 4.3: Changing the Confidence Level

Suppose we change the original problem by using a 95% confidence level. Find a 95% confidence interval for the true (population) mean statistics exam score.

Solution

To find the confidence interval, you need the sample mean,

, and the EBM.

σ =3 ; n = 36 ; The confidence level is 95% (CL = 0.95) CL = 0.95 so α = 1 − CL = 1 − 0.95 = 0.05

The area to the right of z.025 is 0.025 and the area to the left of z.025 is 1−0.025 = 0.975

using invnorm(.975,0,1) on the TI-83,83+,84+ calculators. (This can also be found using appropriate commands on other calculators, using a computer, or using a probability table for the Standard Normal distribution.)

Interpretation

We estimate with 95% confidence that the true population mean for all statistics exam scores is between 67.02 and 68.98.

Explanation of 95% Confidence Level

95% of all confidence intervals constructed in this way contain the true value of the population mean statistics exam score.

Comparing the results

The 90% confidence interval is (67.18, 68.82). The 95% confidence interval is (67.02, 68.98). The 95% confidence interval is wider. If you look at the graphs, because the area 0.95 is larger than the area 0.90, it makes sense that the 95% confidence interval is wider.

Figure 4.1 Comparing the results

Summary: Effect of Changing the Confidence Level

Increasing the confidence level increases the error bound, making the confidence interval wider.
Decreasing the confidence level decreases the error bound, making the confidence interval narrower.

Example 4.4: Changing the Sample Size:

Suppose we change the original problem to see what happens to the error bound if the sample size is changed.

See the following Problem.

Problem

Leave everything the same except the sample size. Use the original 90% confidence level. What happens to the error bound and the confidence interval if we increase the sample size and use n=100 instead of n=36? What happens if we decrease the sample size to n=25 instead of n=36?

σ = 3 ; The confidence level is 90% (CL = 0.90) ;

Solution A
If we increase the sample size n to 100, we decrease the error bound.
When n = 100 :

Solution B

If we decrease the sample size n to 25, we increase the error bound. When n = 25 :

Summary: Effect of Changing the Sample Size

Increasing the sample size causes the error bound to decrease, making the confidence interval narrower.
Decreasing the sample size causes the error bound to increase, making the confidence interval wider.

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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 decreases.

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 does not change 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 increases.

The width of a confidence interval decreases 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

while sampling from a normally distributed population with a standard deviation of

. What is the minimum required sample size?

Possible Answers:

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.

Possible Answers:

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?

Possible Answers:

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?

Possible Answers:

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:

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?

Possible Answers:

Correct answer:

Explanation:

There are

cards in a standard deck:

cards in

suits. There are

face cards (King, Queen, Jack) in each suit, so there are

total face cards.

Thus the probability of choosing a single face card is

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?

Possible Answers:

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?

Possible Answers:

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:

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