AP Statistics – 8.1 Notes Name: __________________ Confidence Intervals: The Basics

Today you will be making your best guess at the mean of the Chapter 7 Test scores by creating a confidence interval (point estimate ± margin of error). The group who creates a confidence interval with the smallest margin of error that still captures the true mean of the Chapter 7 Test score will win a prize.

1. Mr. McGee will give you a random sample of 5 scores. Record the 5 scores and then find the mean. Scores: __________________________________ Mean: ________. This is your point estimate for the true mean.

2. Identify the population, parameter of interest, sample and statistic.

Population: ___________________________ Parameter of Interest: _____________________ Sample:____________________________ Statistic: ____________________

3. Now you are going to change your Point Estimate into an interval of values by adding and subtracting the same number (called Margin of Error) from your point estimate. What margin of error will you choose? Why?

4. Write your interval that you think contains the true mean test score.

5. How confident do you feel that your interval captures the true mean? Answer with a percentage.

6. One of the groups go (71,79) as their interval. What was their point estimate? What was their margin of error?

7. One group claims that the true mean score is 84. Does your interval support or deny this claim? Why?

83 85 90 93 74

I 85

All ch 7 testscores M mean testscore5test scores I 85

PointEstimate 715 75 margin ofError 792712 4

yes bc 84fallswithinmyintervalOR

No bc 84does not fall withinmyinterval

AP Statistics – 8.1 Notes Name: __________________ Confidence Intervals: The Basics

1. Determine the point estimate and margin of error from a confidence level. Practice Problem: In each of the following settings, determine the point estimator you would use and calculate the value of the point estimator: a) The makers of a new golf ball want to estimate the median distance the new balls will travel when

hit by a mechanical driver. They select a random sample of 10 balls and measure the distance each ball travels after being hit by the mechanical driver. Here are the distances (in yards):

285 286 284 285 282 284 287 290 288 285

b) The golf ball manufacturer would also like to investigate the variability of the distance travelled by

the golf balls by estimating the interquartile range. c) The math department wants to know what proportion of its students own a graphing calculator, so

they take a random sample of 100 students and find that 28 own a graphing calculator.

282 284028428512852851 286 0287 288 290PointEstimate

Median_285

PointEstimateIQR Q3 Q1IQR 287 284 IQR 3

f I pointEstimatetoo f 0.28

AP Statistics – 8.1 Notes Name: __________________ Confidence Intervals: The Basics

2. Interpret a confidence interval in context. 3. Interpret a confidence level in context. In this activity you will use the Confidence Intervals applet to learn what it means to say we are “95% confident” that our confidence interval captures the true mean. 1. On the weebly, using the statistical applets button, find the applet called “Statistical Applets:

Confidence Intervals.” Set the confidence level to 95% and the sample size to 5.

2. Click “Sample” to choose an SRS and display the resulting confidence interval. The confidence interval is displayed as a horizontal line segment with a dot representing the sample mean in the middle of the interval. The true (population) mean µ is the green vertical line. a) Did the first confidence interval capture the true mean?

b) Repeat this 10 times and sketch what you see to the right.

3. “Reset” and then take a total of 100 confidence intervals (sample 25 four times). How many out of 100 captured the true mean? Is this surprising? Why?

4. Watch your confidence intervals as you drag the confidence level from 95% to 99% (don’t Reset). What happens to the intervals when the confidence level is increased?

5. “Reset”, then sample 100 times at an 80% confidence interval. How many of the intervals capture the true mean?

6. Now we will see what happens when we adjust the sample size. Change the sample size from 5 o 50 and sample for 1 interval. Then change it to 250 and sample for 1 interval. What happens to the interval when the sample size is increased? Why?

yes Im

AnswersvaryShould be around 95 of intervals

confidence Asyouincrease the confidence level the interval getswidth wider so we hit µ more

AnswersvaryShould be around 80 of intervals

The interval narrows As sample size goesup we expectlessvariability smallerstandarddeviation

AP Statistics – 8.1 Notes Name: __________________ Confidence Intervals: The Basics

4. Interpret a confidence interval in context. 5. Interpret a confidence level in context. 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].” 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.” Interpreting Confidence Levels and Intervals

• 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.

Practice Problems: 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 interval. b) Interpret the confidence level. b) What is the point estimate that was used to create the interval? What is the margin of error? c) 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.

Weare95 Confidentthatthe interval from4,547 to 8,473Capturesthetruemeannumberofsteps taken perdayforemployees

If we tookmanymany samples of size 50 about 95 of theresulting intervals wouldcapturethetrue mean number of stepstakenperday

ME PointEstimate MarginofErrorI 454784732 165,07 8473 6510 119637

4547 I 8473 2

Since all valuesin the interval are lessthan 10,000there is convincing evidencethatemployees are nottaking10,000steps adayon average

AP Statistics – 8.1 Notes Name: __________________ Confidence Intervals: The Basics

6. 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.

Constructing Confidence Intervals Here are two important cautions to keep in mind when constructing and interpreting confidence intervals.

✓ Our method of calculation assumes that the data come from an SRS of size n from the population of interest.

✓ The margin of error in a confidence interval covers only chance variation due to random sampling or random assignment.

Other things being equal, the margin of error of a confidence interval gets smaller as:

1. The confidence level C decreases 2. The sample size n increases.

Practice Problems: A researcher plans to use a random sample of families to estimate the mean monthly family income for a large population.

1. The researcher is deciding between a 95% confidence level and a 99% confidence level. Compared to a 95% confidence interval, a 99% confidence interval will be

(a) narrower and would involve a larger risk of being incorrect. (b) wider and would involve a smaller risk of being incorrect. (c) narrower and would involve a smaller risk of being incorrect. (d) wider and would involve a larger risk of being incorrect. (e) wider and would have the same risk of being incorrect.

2. The researcher is deciding between a sample of size n = 500 and a sample of size n = 1000. Compared to using a sample size of n = 500, a 95% confidence interval based on a sample size of n = 1000 will be

(a) narrower and would involve a larger risk of being incorrect.

(b) wider and would involve a smaller risk of being incorrect. (c) narrower and would involve a smaller risk of being incorrect. (d) wider and would involve a larger risk of being incorrect. (e) narrower and would have the same risk of being incorrect.

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.