CONFIDENCE INTERVALS

NOTES:

Notes

Unbiased Estimator: Unbiased Estimators (mean and proportions) can estimate their population version efficiently through studies of samples.

Point Estimate: The estimation made by an unbiased estimator through a sample. It is the best single guess for a population number through studying a sample.

For proportions, point estimate is denoted as p-hat

For mean, point estimated is denoted as x-bar

Pros and Cons of point estimates: The pro is that it will be close, but the con is that it will almost always be wrong.

To remedy the con (which is pretty bad if you are 99.99999999% wrong), we build a confidence interval.

Confidence Interval: An interval of numbers around the point estimate to better guess the population number.

Margin of Error - denoted as E: An estimate of how wrong the point estimate is based on a confidence level.

Confidence level: The probability the confidence interval correctly guesses the population number

Note that the margin of error and confidence level are directly related. If margin of error goes up, confidence level goes up. If margin of error goes down, confidence level goes down. Another way to explain this is: if you guess more, you have a better chance to guess correctly, similar situation if you guess less. With that said, it would sound like having a lot of margin or error is great if our chances of guessing correctly goes up. But note the name margin oferror.In the end, having too much error is bad. The confidence interval loses value if it contains too much error.

Formula for confidence interval:

(point estimate-E, point estimate+E)

It is important to put the parentheses around the confidence interval. It is written in something called interval notation.

Key Formulas:

alpha (looks like a fish symbol) = 1- confidence level in decimal form

z of alpha over 2 = the positive version of the z-score connected to the probability of alpha/2. If the probability is not in the chart, use the closest one possible.

Common z of alpha over 2 values:

For 90% confidence level: 1.645

For 95% confidence level: 1.96

For 99% confidence level: 2.575

n=sample size

For proportions:

E=(z of alpha over 2)(square root of [(p-hat)(q-hat)/n])

Round to 5 places after the decimal

t-score: Use t-score chart; Relies on area in two tails and degrees of freedom. Using both values, trace down and over.

Area in Two Tails = Alpha = 1-confidence level

degree of freedom=n-1

For mean:

If you only have sample standard deviation:

E=(t-score)(standard deviation)/square root of n

round to 3 places after the decimal

If you have population standard deviation:

E=(z of alpha over 2)(standard deviation)/square root of n

round to 3 places after the decimal

Reminder! Formula for confidence interval:

(point estimate-E, point estimate+E)

Given a confidence interval:

Point estimate = midrange of lower and upper values of confidence interval = (lower + upper)/2

E=(range of upper and lower values)/2=(upper - lower)/2

*no rounding for these

Finding sample size:

For proportions:

n=[(z of alpha over 2)^2]p(1-p)/(E^2)

For mean:

n=[(z over alpha over 2)(standard deviation)/E]^2

n is always roundedupto nearest whole number. The reason why we round up is because if we round down, we won't have enough. The reason why it has to be a whole number is because you cannot have a piece of a sample.

i.e. you cannot interview half a person.

Hypothesis Testing:

A claim is made. You test the claim through a sample. If the test results in an outlier based on the claim, you reject the claim. If the test results in a normal occurance based on the claim, you support the claim.

Make a confidence interval based on the sample. If the claim is within the confidence interval, you support the claim. If the claim is outside the confidence interval, you reject the claim.

1- The researchers want to find the numbers of people they must survey in order to get a better understanding of how many people are aware of climate change. They know from other studies that 91.44% are aware of climate change. They want to build a confidence interval with margin of error that is no more than .00315 and a confidence level of 90%. How many people should the researcher interview?

2-The ages of 4 polar bears in a region were 22, 23, 27, and 29. Build a confidence interval with 95% confidence level for the mean ages of polar bears.

What is theuppervalue of the confidence interval?

3- A point estimate is the best-good guess for a population number based on a sample. The confidence interval also tries to guess the population number. If we already have the best guess, the point estimate, then why do we need to do more work to make a confidence interval?

A-It is not even a good guess

B-There is nothing wrong with it

C-It is always wrong

D-It is almost always wrong

4-Based on particular sample, which confidence interval is wider: the 95% confidence interval or the 80% confidence interval?

A-80% is wider

B-95% is wider