6.1 CI

Critical Values & Constructing Confidence Intervals for 1 population proportion

Exploring Z-scores to use in Confidence intervals

Directions:Recall that to compute a 95% confidence interval estimate of a population value, we use approximately 2 standard deviation from the center.Recall we get the two standard deviations from the Empirical Rule.But the Empirical rule says that 95% of the data will fall within "approximately" two standard deviations.But how accurate is 2 standard deviations if it says "approximately"?Can we find a more accurate answer to the number of standard deviations from the center that 95% of the data is in between?What about 90% or 99% since those are also commonly used?These values are Z-scores and are often called "Critical Values" in Statistics.

1.Use Statcrunch to find the two Z-scores that corresponds to the middle 95%.Draw a picture showing the Z-scores and 95%.Remember that the area under the curve between these Z-scores must be 0.95.Do you remember what mean and standard deviation we use to find Z-scores on Statcrunch?These Z-scores are pretty famous and are the Z-scores for 95% confidence intervals .

2.If we use a 99% confidence level instead of 95% do you think the Z-score will be more or less than + or - 2?Draw a picture and explain why you think so.Now use Statcrunch to find the two Z-scores that we could use to calculate a 99% confidence interval.How well did your first guess agree with what we found on Statcrunch?

3.Now repeat #2, but use a 90% confidence interval.If we use a 90% confidence level instead of 95% do you think the Z-score will be more or less than + or - 2?Draw a picture and explain why you think so.Now use Statcrunch to find the two Z-scores that we could use to calculate a 90% confidence interval.How well did your first guess agree with what we found on Statcrunch?

Let's Summarize the Z-score critical values that we found.These are important to memorize as we will be using them constantly in inferential statistics.

Confidence LevelZ score for the confidence interval

90%

95%

99%

Confidence intervals give two values that we think the population value is in between.To construct a confidence interval, we start with the sample value (point estimate) and then add and subtract a certain number of standard deviations from the sample value.These standard deviations are also called standard errors.The number of standard errors is the critical z-scores corresponding to a certain confidence level.

Directions:For numbers 4-6, use the formula and your calculator to calculate the confidence interval estimate of the population percentp. You may have to use the formula to calculate the sample percent.Also remember to write the sample proportion as a decimal before plugging into the formula.

4. In a random sample of 72 adults in Santa Clarita, CA, each person was asked if they support the death penalty.31 adults in the sample said that they do support the death penalty.What was the sample proportion of adults in Santa Clarita that support the death penalty?Now calculate a 95% confidence interval population estimate of people in Santa Clarita that support the death penalty. Remember to use the appropriate critical value Z-score for each.

5. In a random sample of 400 Americans, each person was asked if they are satisfied with the amount of vacation time they given by their employers. 84% of them said that they were not satisfied with their vacation time.Calculate the following.What was the sample proportion of Americans that were not satisfied with their vacation time?Now construct a 99% confidence interval in order to estimate the percent of Americans that are not satisfied with their vacation time.

6. What percent of eligible Americans vote?In 2008, a random sample of 3000 American adults that were eligible to vote was taken and we found that 2040 of them voted.Construct a 90% confidence interval estimate of the population percent of Americans that vote.Now construct another confidence interval.This time construct a 90% confidence interval estimate of the population percent of Americans that do not vote.Hint:For the "do not vote" group, the sample percent will change.