CONFIDENCE STATEMENT MARGIN OF ERROR CONFIDENCE INTERVAL 1

Class Objective After this class, you will be able to - Determine and interpret a Confidence Statement 2

Terminologies and symbol Sample proportion: p (p-hat) – one type of statistic Example: n (sample size) = out of n say “No” p = 25% Population proportion: P – one type of parameter We use p-hat to infer P

Class Activity: Draw Samples of different sample size Assume there are only 50 residents in CT. You want to find out the average number of how many times CT residents go to movie per month. Use the first two rows of number of your random number table to simulate the response of the 50 residents. Use the sample size n=1, n=5, n= 25 to find the range of the sample means for different sample sizes.

Class Work What can you conclude from the class activity –Draw Samples of different sizes 5

Conclusion Larger the sample size – smaller the error made in inference Always use a larger sample size whenever it is possible

Confidence Statements Definition: a fact about what happens in all possible samples Say how much we can trust the result of one sample Has 2 parts: Margin of error: how close the (sample) statistic lies to the (population) parameter Level of confidence: what percentage of all possible samples satisfy the margin of error 7

Confidence Statements - Interpretation Chris‘ research showed that 22% of the U.S. high school students spent 49 hours a week on texting. For 95% confidence, the margin of error is 4%. Therefore: (Interpretation) Between 18% and 26% of all U.S. high school students spent 49 hours a week on texting It means that Chris’ sample might be one of the 5% that miss by more than 4% The interval 18% to 22% may or may not capture the percent of U.S. high school students who spent 49 hours a week on texting. But, in the long run this procedure will produce intervals that capture the unknown population values (P) about 95% of the time (Probability) => called the 95% confidence level. 8

9 The Importance of Religion for Adult Americans Poll of n = 1025 adult Americans: “How important would you say religion is in your own life?” Very important56% Fairly important25% Not very important19% Margin of error is 3%: NOW, you try to interpret the findings of subject survey.

10 The Importance of Religion for Adult Americans Approx. 95% confidence interval for the percent of all adult Americans who say religion is very important: 56%  3% or 53% to 59% The interval 53% to 59% may or may not capture the percent of adult Americans who considered religion to be very important in their lives. But, in the long run this procedure will produce intervals that capture the unknown population values about 95% of the time => called the 95% confidence level.

Class Work So we know how to interpret a confidence statement. What will be the next question regarding a confidence statement? 11

12 Margin of Error: The Accuracy of Sample Surveys The sample proportion and the population proportion with a certain trait or opinion differ by less than the margin of error in at least 95% of all random samples. Conservative margin of error = Add and subtract the margin of error to create an approximate 95% confidence interval.

13 Confidence Intervals 95% Confidence Interval for a Population Proportion: For about 95% of properly conducted sample surveys, the interval sample proportion to sample proportion will contain the actual population proportion. Another way to write it: sample proportion

A “Quick” method for Margin of Error 14

Now you try it In December 2012, the Callup Poll asked a random sample of 1007 adults, “Thinking of your own situation, do you personally own a cat, a dog, both, or neither?” It is found that 59% said they owned a cat, a dog, or both. Write a confidence statement and interpret it! 15

Class Work For the same sample, a 99% confidence statement would have a larger or smaller (which one is the answer) margin of error than a 95% confidence statement. Explain why. 16

Confidence Statements For the same sample, a 99% confidence statement will have a larger margin of error than a 95% confidence statement 17

Class Work What would be the relationship between sample size and margin of error? 18

19 Choosing a Sample Size for a Survey If m.e. is the desired margin of error for a 95% confidence interval for a population proportion, the required sample size is:

Confidence Statements To have a smaller margin of error (means smaller variability) a larger sample size is needed ( 1/ square root of n) 20

Confidence Statements The conclusion of a confidence statement applies to the population, not to the sample. The conclusion about the population is never completely certain 21

22 The Effect of Population Size The m.e. for a sample of 1000 is about 3% whether the population size is 30,000 or 200 million. (inferential statistics – we use sample to infer the population) In practice, as long as the population size is ≥ 10 times as large as the sample size, the population size has almost no influence on the accuracy of sample estimates.

Homework Assignment: Chapter 5 Exercise 5.25 and 5.30 Reading: Chapter 5 – p