MATH 1107Elementary Statistics

Lecture 8Confidence Intervals for

proportions and parameters

All around Atlanta, we are increasingly seeing cameras at intersections – a low cost way to ticket people for running lights. Its an election year. A local representative wants to know what people think about the new cameras.Lets say that 829 people were surveyed regarding the “photo-cop” and 423 said they hated it. How would you report the results? What is the representative figure?

MATH 1107 – Confidence Intervals

Definition: A confidence interval (or interval estimate) is a range (or an interval) of values used to estimate the true value of a population parameter. A confidence interval is sometimes abbreviated as CI.

A confidence level is the probability 1— (often expressed as the equivalent percentage value) that is the proportion of times that the confidence interval actually does contain the population parameter, assuming that the estimation process is repeated a large number of times.

This is usually 90%, 95%, or 99%.

( = 10%), ( = 5%), ( = 1%)

MATH 1107 – Confidence Intervals

In the example, we noted that 829 adults were surveyed, and 51% of them were opposed to the use of the photo-cop for issuing traffic tickets. Using these survey results, find the 95% confidence interval of the proportion of all Atlantans opposed to photo-cop use.

“We are 95% confident that the interval from 0.476 to 0.544 does contain the true value of p.”

How was this calculated?

Confidence Interval Estimation

MATH 1107 – Confidence Intervals

p z /2 * [(pq)/n]

Where:

p = population proportionz /2 = the appropriate Z-score based upon the selected value q = 1-pn = number of elements in sample

Lets take a look at the formula that we would use to answer this question (page 429):

So, how did we calculate“We are 95% confident that the interval from 0.476 to 0.544 does contain the true value of p.”?

Here, p=.51q=.49z=1.96n=829

So,.51+ 1.96*(SQRT(.51*.49)/829)

.51+ .034 or .476 to .544

(.51-.034 = .476 and .51+.034 = .544)

MATH 1107 – Confidence Intervals

Where in the world did Z=1.96 come from?

MATH 1107 – Confidence Intervals

What if I wanted to be 90% confident?What if I wanted to be 99% confident?

Typical Z scores used in CI Estimation:90% confidence = 1.64595% confidence = 1.9699% confidence = 2.575

MATH 1107 – Confidence Intervals

MATH 1107 – Confidence Intervals

Go to http://www.gallup.com/

Lets replicate the findings from the Gallup poll of the day.

MATH 1107 – Confidence Intervals

Question – what is the relationship between the Margin of Error and the confidence level?

If we need to be more confident, what happens to the Margin of Error? What if we need to maintain the MOE, AND increase the level of confidence?

x z /2 * (s/SQRT(n))

Where:

x = sample meanz /2 = the appropriate Z-scores = sample standard deviationn = number of elements in sample

Conceptually, we use the same process for estimating the confidence interval of a parameter:

-

MATH 1107 – Confidence Intervals

MATH 1107 – Confidence Intervals

Lets say that we took a poll of 100 KSU students and determined that they spent an average of $225 on books in a semester with a std dev of $50.

Report the 95% confidence interval for the expenditure on books for ALL KSU students.

Now, assuming that you need to maintain this MOE, but at a 99% confidence, what is the new sample size?

MATH 1107 – Confidence Intervals

Fun EXCEL exercise !