AP Statistics – Ch 8.1 Name: ________________ Confidence Intervals: The Basics

Suppose we are interested in finding out “How many pairs of shoes a Liberty High female teen has, on average” • What is the population and the parameter we are looking to find?

– Population: LHS Females – Parameter: ______________________________

• Let’s also suppose we know the population standard deviation is 15. – μ = ??? – σ = 15.5

• To answer this question we took an SRS of 30 students • Here is an SRS (n = 30) of Liberty High females and the number of shoes they own.

50 26 26 31 57 19 24 22 23 38

31 29 15 45 40 20 31 29 25 46

13 50 13 34 23 30 49 13 15 51

x = ____ If you had to give one number to estimate an unknown population parameter, what would it be? If you were estimating a population mean P, you would probably use �̅�. If you were estimating a population proportion p, you might use �̂�. In both cases, you would be providing a Point Estimate of the parameter of interest.

Practice Problem: In each of the following settings, determine the point estimator you would use and calculate the value of the point estimator: 1. 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

2. The golf ball manufacturer would also like to investigate the variability of the distance travelled by

the golf balls by estimating the interquartile range. 3. 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.

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.

Mmean ofshoesforLHSfemale

30.6

282 284028428512852851 286 0287 288 290PointEstimate

Median_285

PointEstimateIQR Q3 Q1IQR 287 284 IQR 3

f I pointEstimate100 pi 0.28

AP Statistics – Ch 8.1 Name: ________________ Confidence Intervals: The Basics

LHS Female’s Shoes • Is the value of the population mean μ, the average number of shoes owned by LHS females, exactly

equal to 30.6? – No, we can only guess that its somewhere around 30.6

• How close is it? • To answer this, we ask another question: “How would the sample mean x vary if we took many

SRSs of size n = 30 from this same population?”

• If we took ALL possible sample size n = 30 from the population, this would be called the…. __________________________

Sampling Distribution of x

• Shape: Since our sample size is greater than 30 it will be Normal (CLT) • Center: The mean of the sampling distribution of x is the same as the unknown mean μ of the

entire population. That is PP x • Spread: the standard deviation of the sampling distribution of x for sample sizes n = 30 is (30

students is less than 10% of all students)

The Idea of a Confidence Interval 1. The sample mean x = 30.6 is our point estimate, but It isn’t exactly equal to the unknown μ. 2. In repeated samples (n = 30), the values of x follow a Normal distribution with mean μ and

standard deviation of 2.8 3. The 95 part of the 68 – 95 – 99.7 Rule for Normal distributions says that x is within 2 (2.8) = 5.6

(That’s 2 standard deviations) of the population mean μ in about 95% of all samples of size n = 30 4. Whenever x is within 5.6 points of μ, μ is within 5.6 points of x . This happens in about 95% of

samples. So the interval �̅� − 5.6 to �̅� + 5.6 5. If we estimate that μ lies somewhere in the interval from

�̅� − 5.6 = 30.6 − 5.6 = 𝟐𝟓 �̅� + 5.6 = 30.6 + 5.6 = 36.2

We’d be calculating this interval using a method that captures the true μ in about 95% of all possible samples of this size.

The Big Idea: The sampling distribution of x tells us how close to μ the sample mean x is likely to be. All confidence intervals we construct will have form similar to this:

_____________________________

xP 8.2�xP 6.5�xP 4.8�xP8.2�xP6.5�xP4.8�xP

SamplingDistributionof

Of 157 2.8

Estimate I MarginofError

Interpreting Confidence IntervalWe are 95 Confidentthattheintervalfrom 25to 36.2Captures thetruemeanvalue ofshoesownedbyLHSfemales

AP Statistics – Ch 8.1 Name: ________________ Confidence Intervals: The Basics

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.”

Practice Problem 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) 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.

Interpreting Confidence Levels and Intervals The confidence level does not tell us _____________________________ that a particular confidence interval captures the population parameter. Instead, the confidence interval gives us a set of _______________________ values for the parameter. Constructing Confidence Intervals

A ____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.

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.

C

Interval Level

weare95 Confidentthatthe interval from4,547 to 8,473Capturesthetruemeannumberof steps taken perdayforemployees

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 a dayon average

He chance

plausible

AP Statistics – Ch 8.1 Name: ________________ Confidence Intervals: The Basics

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.

Practice Problem: According to the American Community Survey, a 95% confidence interval for the median household income in Texas during the years 2009–2011 is $58,929 ± $218.

Problem: Interpret the confidence interval and the confidence level.

Key

ConfidenceInterval We are 95 Confident that theintervalfrom58,711 to 59 147 Captures thetrue medianincomefor households in TX

fenfidenee Level If ManySamples of TXhouseholdsweretaken about95of theresultingintervals wouldcapturethetruemedianincomeof TX households