Chapter 8: Estimating with Confidence Part IV: Inference Chapter 8: Estimating with Confidence

8.1: Confidence Intervals – The Basics

Definitions A point estimator is a statistic that provides an estimate of a population parameter. The value of that statistic is called a point estimate. Ideally, a point estimate is our “best guess” at the value of an unknown parameter.

Conversation Hearts Your group has been given a bag of hearts. Select a sample of 5 hearts. What proportion of the hearts are pink? Determine a point estimate for the proportion of pink hearts in the bag.

Just like our classroom.  Conversation Hearts   Just like our classroom. 

Conversation Hearts Repeat this process for a sample of… Now find the true proportion of pink hearts in the bag. Did any of your samples capture the true proportion?

What You Already Know…I Hope Suppose we are interested in the proportion of left-handed professional baseball players. In our sample of 59 MLB players, 15 are left- handed.   Find the point-estimate. What interval will the true proportion of left- handed MLB players will contain 95% of all samples of size 59? 68%? 99.7?

The idea of a confidence interval  

Interpreting Confidence Intervals Confidence Interval Applet Examine what happens with many intervals…

Interpreting confidence intervals Confidence level: To say that we are 95% confident is shorthand for “95% of all possible samples of a given size from this population will result in an interval that captures the unknown parameter.” Confidence interval: 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].”

Interpreting Confidence intervals For example, In late 2009, the Pew Internet and American Life Project asked a random sample of 2253 U.S. adults, “Do you ever…use Twitter or another service to share updates about yourself or to see updates about others?” Of the sample, 19% said “Yes.” According to Pew, the resulting 95% confidence interval is (0.167, 0.213). Confidence Interval: We are 95% confident that the interval from 0.167 to 0.213 contains the true proportion of U.S. adults who use Twitter or another service for updates. Confidence Level: In 95% of all possible samples of 2253 U.S. adults, the resulting confidence interval would capture the actual population proportion of U.S. adults who use Twitter or another service for updates.

Keep in mind… What’s the probability that our 95% confidence interval captures the parameter? Hint: It’s NOT 95%! (that is our level of certainty or confidence)

Constructing a confidence interval Confidence Interval Applet Consider the relationship between the confidence level and the confidence interval.

Constructing a confidence interval   margin of error

Properties of confidence intervals YOU choose the confidence level – the critical value depends on the level you choose. Greater confidence requires a larger critical value. The standard deviation of the statistic still depends on the sample size n (as we saw in Chapter 7). The margin of error gets smaller when: The confidence level decreases The sample size n increases

Conditions for constructing a confidence interval Random: The data come from a well- designed random sample or randomized experiment. Normal: For means – stated, or CLT (based on sample size) If you are given the data, you can also visually inspect it to see if it looks approx. normal Independent: addressed with the 10% condition

Homework Pg. 482: #5 – 13 odd, 17, 19 – 24 all, 27, 31, 33