8 Confidence Intervals

Question 1

What is the main point of making a confidence interval?

Answer 1

When we want to estimate an unknown population parameter (usually a mean or a proportion).

Question 2

In a confidence interval, what is the margin of error?

Answer 2

It is how much we add and subtract to our statistic (point estimate) to make an interval (range of values). It is half the width of the interval. It is created by multiplying the standard error (deviation) by the critical value that goes with our confidence level.

Question 3

What will decrease the width of a confidence interval? OR What will decrease the margin of error?

Answer 3

If you lower the confidence level, the critical value will be smaller so the interval will be narrower. If you increase your sample size, the standard error (spread) reduce and so will your C.I. width.

Question 4

How do we interpret a confidence interval?

Answer 4

We are C% confident (sure) that the interval ( , ) captures the true (mean/proportion)…CONTEXT. Basically: we’re pretty sure the real answer is somewhere between the two numbers in our interval.

Question 5

What do you do for the P step of PANIC?

Answer 5

Parameter: “We want to estimate the true proportion of CONTEXT at a C% level.”

Question 6

What do you do for the A step of PANIC?

Answer 6

Assumptions: Check the three conditions: Random, Normal (large counts), and Independent (10% condition).

Question 7

What do you do for the N step of PANIC?

Answer 7

Name the test/procedure you are using: 1 (or 2) sample z-interval for p.

Question 8

What do you do for the I step of PANIC?

Answer 8

Interval: List your interval: ( , ). As long as you name your procedure, you don’t have to show any work.

Question 9

What do you do for the C step of PANIC?

Answer 9

Conclude: We are C% confident (sure) that the interval ( , ) captures the true (mean/proportion)…CONTEXT.

Question 10

How do you find the critical value for a confidence interval?

Answer 10

It’s based on the C% level. Invnorm of C% (as a decimal) and use the center tail. If you don’t have center tail as an option, then invnorm(tail). Example: 90 has 10 on the tails so one tail is 5 so invnorm (.05).

Question 11

What is the critical value (z-score) for 95%?

Answer 11

1.96

Question 12

How do you check the normal (large counts) condition for proportion problems?

Answer 12

State the number of successes and failures are both at least 10. Example: 30 out 80 people agree with something. You write 30 and 50 (total: 80 minus successes (30)) are both greater than 10. If you don’t know the number of successes but instead know p-hat, do np and n(1-p) to find successes and failures.

Question 13

How do you check the independent condition?

Answer 13

n (sample size) < 10% of all CONTEXT. In a PANIC problem you can just state this. In a different problem, you might have to make the argument that this condition is not met. Example: They collected data from 600 students but it is doubtful that there are more than 6000 students in this high school.

Question 14

What does it mean to say we are 95% confident? i.e. Interpret the confidence level.

Answer 14

If we repeat this process/method (taking a sample and creating a confidence interval) many, many times, it will work (capture the true parameter) about 95% of the time.

Question 15

Given a confidence interval, how do you find the point estimate and margin of error?

Answer 15

The point estimate is the middle number and the MoE is half the width. Example: CI: (.22 , .34). The point estimate is .22+.34= .56/2 = .28. The MoE is half of the width: .34-.22 = .12/2 = .06.

Question 16

How do you use a confidence interval to test a claim?

Answer 16

If the claim is not in the interval, then you have good evidence to reject the claim. Otherwise, you have evidence that supports the claim. Example: “A majority of people will support this new law” but when you did a study, your CI was (.31, .39). Since .50 is not in my interval I have good evidence to reject the claim.

Question 17

How do you do a confidence interval on a calculator?

Answer 17

STAT, TESTS, 1 (or 2) Prop Z int. You have to input the successes as a whole number. So if you only have p-hat you need to find np and round it.

Question 18

How do you find a sample size given a margin of error?

Answer 18

Set up the formula and solve for n: MoE = Z * Sqrt(p(1-p)/n). 3 steps: divide by Z, square both sides, switch the n and the other side of the equation. Always round up to the next whole person.

Question 19

On the formula sheet, a confidence interval is: statistic plus/minus (critical value)*(standard error of statistic). How do you find these 3 numbers?

Answer 19

Statistic: p-hat given in problem or success/total given in problem.
Critical value: if 95%, 1.96, otherwise, use invnorm(tail).
Standard error of statistic: pick the correct formula from the back of the formula sheet.

Question 20

For a 2-sample interval, how do you know if there is a difference in proportions between the 2 samples?

Answer 20

If 0 is not in your interval, then you have good evidence that the true proportion is higher for one of the groups than the other. If 0 is in the interval, there is not convincing evidence of a difference in population proportions.