8 1 Confidence Intervals The Basics Objectives SWBAT

8. 1 Confidence Intervals: The Basics Objectives SWBAT: • DETERMINE the point estimate and margin of error from a confidence interval. • INTERPRET a confidence interval in context. • INTERPRET a confidence level in context. • DESCRIBE how the sample size and confidence level affect the length of a confidence interval. • EXPLAIN how practical issues like nonresponse, undercoverage, and response bias can affect the interpretation of a confidence interval.

How is this chapter different than Chapter 7? Chapter 8 is like the time that 7 strangers were picked together to live under one roof, and have their lives taped, to find out what happens when people stop being polite, and start getting real…Real World, AP Statistics.

How confident would you be in saying that it will be exactly 35 degrees tomorrow? Would you be more or less confident if you came up with a range of values? • For example, how about if you said it would be between 30 and 40 degrees? • What about if you said it would be between 20 and 50 degrees? How confident would you be in this prediction? • What would the problem be with coming up with too large of a range? • Obviously we would be pretty confident in this interval. However, it doesn’t really tell us anything about the weather. • We don’t know if it will be hot or cold (only Katy Perry knows that).

What is a point estimate? Why is it called a point estimate? A point estimator is a statistic that provides an estimate of a population parameter. The value of that statistic from a sample is called a point estimate. • The point estimate is our single best guess for the value of a population parameter (think about one single value on the number line). However, it almost certainly is not correct? • Therefore, what’s our solution? • Use an interval of values!

Explain the logic of confidence intervals. Let’s say that from a population, we take a sample of size 16 that has a mean of 240. 79 and a standard deviation of 5. If we estimate that µ lies somewhere in the interval 230. 79 to 250. 79, we’d be calculating an interval using a method that captures the true µ in about 95% of all possible samples of this size.

What is a confidence interval? estimate ± margin of error A C% confidence interval gives an interval of plausible values for a parameter. The interval is calculated from the data and has the form point estimate ± margin of error The difference between the point estimate and the true parameter value will be less than the margin of error in C% of all samples. The confidence level C gives the overall success rate of the method for calculating the confidence interval. That is, in C% of all possible samples, the method would yield an interval that captures the true parameter value.

What is the margin of error? Why do we include the margin of error? • Margin of error is how far we expect the sample statistic to vary from the population parameter, AT MOST. • We include margin of error to account for sampling variability – the fact that different samples will give us different estimates. • We do not include the margin of error to account for bias! • Additionally, include a margin of error greatly increases our changes of being correct. Think about the Garfield comic, but to a smaller degree.

How do you interpret a confidence interval? Interpreting Confidence Intervals To interpret a C% confidence interval for an unknown parameter, say, “We are C% confident that the interval from _____ to _____ captures the actual value of the [population parameter in context]. ” So if constructing a 95% confidence interval, your interpretation would be: “We are 95% confident that the interval from to contains the true. ”

We are 95% confident that the interval from 59% to 67% contains the true proportion of American adults who support the death penalty.

How do you interpret a confidence level? In other words, what does it mean to be 95% confident? • Confidence Intervals Applet Interpreting Confidence Levels To say that we are 95% confident is shorthand for “If we take many samples of the same size from this population, about 95% of them will result in an interval that captures the actual parameter value. ” The confidence level tells us how likely it is that the method we are using will produce an interval that captures the population parameter if we use it many times. The confidence level does not tell us the chance that a particular confidence interval captures the population parameter. Instead, the confidence interval gives us a set of plausible values for the parameter. We interpret confidence levels and confidence intervals in much the same way whether we are estimating a population mean, proportion, or some other parameter.

Alternate Example: A large company is concerned that many of its employees are in poor physical condition, which can result in decreased productivity. To determine how many steps each employee takes per day, on average, the company provides a pedometer to 50 randomly selected employees to use for one 24 -hour period. After collecting the data, the company statistician reports a 95% confidence interval of 4547 steps to 8473 steps. a) Interpret the confidence level. If we were to take many, many samples and calculate many, many intervals, about 95% of the intervals will capture the true mean number of steps taken per day for employees at this company. b) Interpret the confidence interval. We are 95% confident that the interval from 4547 to 8473 captures the true mean number of steps taken per day for employees at this company. c) What is the point estimate that was used to create the interval? The margin of error? The point estimate is the midpoint of the interval: (4547+8473)/2= 6510. The margin of error is the distance from the point estimate to each endpoint of the interval: 8473 – 6510 = 1963. d) Recent guidelines suggest that people aim for 10, 000 steps per day. Is there convincing evidence that the employees of this company are not meeting the guideline, on average? Explain. Because all the values in the interval are less than 10, 000, there is convincing evidence that the employees of this company are not taking 10, 000 steps per day, on average. That is, there is convincing evidence that the employees are not meeting the guideline.

What is the formula for calculating a confidence interval? Is this formula included on the formula sheet? Calculating a Confidence Interval The confidence interval for estimating a population parameter has the form statistic ± (critical value) • (standard deviation of statistic) where the statistic we use is the point estimator for the parameter. Formula sheet! Boom! Prescription for happiness:

How can we reduce the margin of error in a confidence interval? Why do we want a small margin of error? Are there any drawbacks to these actions? • One way to reduce the margin of error is to reduce the confidence level. The drawback to this is you are less likely to get a correct estimate. • Another way to reduce the margin of error is to increase the sample size. The drawback to this is that when we are physically collecting the sample, a larger sample size will be more costly. • The benefit of a small margin of error is that we are more precise.

What are two important things to remember when constructing and interpreting confidence intervals? ü Our method of calculation assumes that the data come from an SRS of size n from the population of interest. ü The margin of error in a confidence interval covers only chance variation due to random sampling or random assignment.

In a 2009 survey, researchers asked random samples of US teens and adults if they use social networking sites. Overall, 73% of the teens said yes and 47% of the adults said yes. A 90% confidence interval for the true difference in the proportion of teens and adults who would say yes is 0. 229 to 0. 291. a) Interpret the confidence level. If we were to take many, many samples and calculate many, many intervals, about 90% of the intervals will capture the true difference in the proportion of teens and adults who use social networking sites. b) Interpret the confidence interval. We are 90% confident that the interval from 0. 229 to 0. 291 captures the true difference in the proportion of teens and adults who use social networking sites. c) Based on the interval, is there convincing evidence that the proportion of teens who would say yes is higher than the proportion of adults who would say yes? Explain. Yes because 0 is not contained in the interval. Every value in the interval has the proportion of teens saying yes higher than the proportion of adults saying yes. d) How would the interval be affected if we used a 99% confidence level instead of a 90% confidence level? The interval would be larger in order for us to be more confident.