CONFIDENCE INTERVALS: THE BASICS

Unit 8 Lesson 1

Estimating with Confidence

How long does a new model of a laptop battery last?

It wouldn’t be practical to determine the lifetime of every laptop battery. Instead, we choose of sample of batteries to represent the population and collect data.

This is a sample statistic to estimate an unknown population parameter.

Statistical Inference provides methods for drawing conclusions about a population from sample data.

Estimating with Confidence

We have, of course, been drawing conclusions from data all along.

What is new is formal inference uses the language of probability to express the strength of our conclusions.

Activity – the Mystery M

Before class, your teacher stored a value for µ (represented by M) in the calculator.

By using the calculator, we will choose an SRS of 16 observations from a Normal population with mean M and standard deviation of 20, and then compute the mean of those 16 sample values.

Your turn! Your group must determine an interval of reasonable values for the population mean µ. Think about the value that the calculator gave us and what you learned about sampling distributions in the previous chapter.

On the sheet of paper, largely and neatly write your team’s results (remember, you’re writing an interval) and put a clear, short, concise of how you reached that interval.

Activity – the Mystery M

Here is a summary of the calculator output: The population distribution is _____________ and its

standard deviation is ___________. A simple random sample of n = _____ observations

was taken from this population The calculator output gave us the sample ______, and

the notation for this is ______.If we had to give a single number to estimate

the value of M that your teacher chose, what would it be? _____ This is known as a point estimate.

We are using the statistic as a point estimator of the parameter µ.

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x

Normalσ = 20

16

mean

Vocabulary

Point estimator = a statistic that provides an estimate of a population parameter.

Point estimate = the value of that statistic from a sample. This is our “best guess” of the value of an unknown parameter.

EXAMPLE

In each of the following settings, determine the point estimator you would use and calculate the value of the point estimate (use the correct notation). 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 285Use the sample median as a point estimator for the

true median. The sample median is 285 yards.

EXAMPLE

In each of the following settings, determine the point estimator you would use and calculate the value of the point estimate (use the correct notation). 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.

Use the sample IQR as a point estimator for the true IQR. The sample IQR is 287 – 284 = 3 yards

EXAMPLE

In each of the following settings, determine the point estimator you would use and calculate the value of the point estimate (use the correct notation). 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.

Use the sample proportion as a point estimator for the true proportion p. The sample proportion is =0.28

p̂

p̂

YOUR TURN!!!

A) Quality control inspectors want to estimate the mean lifetime µ of the AA batteries produced in an hour at a factory. They select a random sample of 30 batteries during each hour of production and then drain them under conditions that mimic normal use. Here are the lifetimes (in hours) of the batteries from one such sample.

16.91 18.83 17.58 15.84 17.42 17.65 16.63 16.84 15.63

16.37 15.80 15.93 15.81 17.45 16.85 16.33 16.22 16.59

17.13 17.10 16.96 16.40 17.35 16.37 15.98 16.52 17.04

17.07 15.73 16.74

The use of the sample mean as a point estimator for the population mean µ. For these data, our estimate is =16.7 hours.

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x

YOUR TURN!!!

B) What proportion p of U.S. high school students smoke? The 2007 Youth Risk Behavioral Survey questioned a random sample of 14,041 students in grade 9 to 12. Of these, 2808 said they had smoked cigarettes at least one day in the past month.

Use the sample proportion as a point estimator for the population proportion p. For this survey, our point estimate is 2808/14,041 = 0.20.

p̂

p̂

YOUR TURN!!!

C) The quality control inspectors in part (A) want to investigate the variability in battery lifetimes by estimating the population variance σ2.

Use the sample variance as a point estimator for the population variance σ2 . For the battery life data, our point estimate is = 0.508.

2xs

2xs

The Idea of a Confidence Interval – Mystery M

Do you think the value the calculator gave us is the exact value that your teacher put in the calculator? Probably not.

This is an estimate. We guess that µ is “somewhere around” this value.

How would the sample mean vary if we took many SRSs of size 16 from this same population?

The sampling distribution of describes how the values of vary in repeated samples. (Recall the facts about this sampling distribution from last unit)

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Review of last unit: sampling distributions

Review of sampling distributions: If population distribution is Normal, then is Normal.

If population distribution is NOT Normal, then is Normal if n ≥ 30.

(as long as n≤10%N or 10n≤N)

Since we could also use inference for proportions, let’s review sampling distributions too: To check if Normal, must follow np≥10 and n(1-p)≥10 (as long as n≤10%N or 10n≤N)

x

p̂

x

x x

nx

pp ˆ

n

ppp

)1(ˆ

Back to the Mystery M (with sampling distribution)

Shape: Because the population is Normal, so is the sampling distribution of .

Center: The mean of the sampling distribution of is the same as the unknown µ of the entire population.

Spread: (the 10% condition is met – we are sampling from an infinite population in this case. )

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x

x

516

20

nx

Moving beyond a point estimate – to the interval

Draw the sampling distribution of .Remember, the Empirical Rule 68-95-99.7. In 95% of all samples, lies within ±10 of the

unknown population mean µ. So µ lies within ±10 of in those samples.

So, with our Mystery Mean problem, we can say that __________________ gives an approximate 95% confidence interval for µ.

This interval can also be written in other ways: ________ to ________ ________≤ µ ≤ _________ (_______, _______)

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Vocabulary

The estimate (in this example, = _____ ) is our best guess for the value of the unknown parameter.

The margin of error, (in this example, 10) shows how close we believe our guess is, based on the variability of the estimate in repeated SRSs of size 16.

The confidence interval is about 95% because the interval (in this example, ) catches the unknown parameter in about 95% of all possible samples. Confidence intervals can also be referred to as interval

estimates.

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10x

VOCABULARY

A confidence interval for a parameter has 2 parts: An interval calculated from the data, which has the form

estimate ± margin of error The margin of error tells how close the estimate tends to be

to the unknown parameter in repeated sampling.A confidence level, C, which 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. We usually choose a confidence level of 90% or higher

because we want to be quite sure of our conclusions. The most common confidence level is 95%.

Applet

To visually display many confidence intervals: www.whfreeman.com/tps4e

http://bcs.whfreeman.com/tps4e/#628644__666391__

Let’s do the same with our Mystery M

Let’s take many (let’s say 25) SRSs of 16 observations and record the results similar to the applet.

Here’s what you should notice: The center of each interval is marked by a dot. The distance from the dot to either endpoint of the

interval is the margin of error. ____ of these 25 intervals (that’s ____) contain the true

value of µ.

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Interpreting Confidence Levels and Confidence Intervals

Confidence level: To say that we are 95% confident is shorthand for saying “95% of all possible samples of a given size from the 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]

Note: always be sure to interpret confidence intervals, only interpret confidence levels if prompted to do so.

Back to the Mystery M problem…

Confidence level: The confidence level in the mystery mean example – roughly 95% - tells us that in about 95% of all SRSs of size 16 from my teacher’s mystery population, the interval will contain the population mean µ.

Confidence interval: We are about 95% confident that the interval from ______ to _____ captures the mystery mean.

There are only 2 bases for the confidence:1)The interval contains the mean, or 2)The interval doesn’t contain the mean.

10x

EXAMPLE

According to the Gallup poll, on August 13, 2010, the 95% confidence interval for the true proportion of Americans who approved of the job Barack Obama was doing as president was 0.44 ± 0.03.

Interpret the confidence interval and the confidence level.Confidence interval: We are 95% confident that the

interval from 0.41 to 0.47 captures the true proportion of Americans who approve of the job Barack Obama was doing as president at the time of the poll.

Confidence level: In 95% of all possible samples of the same size, the resulting confidence interval would capture the true proportion of Americans who approve the job Barack Obama was doing as president.

For more examples, reference your book on p. 476.

Be careful about confidence intervals…

The confidence interval 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.

We tend to calculate only a single confidence interval for a given situation – this level does NOT tell us the probability of that one particular interval capturing the population parameter.

We have a 95% chance of getting a sample mean that’s within 2 standard deviations of the mystery µ - that means our confidence interval captures the µ.

This does NOT mean that there is a 95% chance that our resulting 95% confidence interval captures the µ - after we create a confidence interval, it either captures the µ (100%) or it doesn’t (0%).

Constructing a Confidence Interval

Refer back to the applet.Set the confidence interval to 80% and click

“Sample 50”. Now change to 90%, notice the percent hit.Now change to 95%, notice the percent hit.Now change to 99%, notice the percent hit.Click on them several times to visually see

the intervals and the percent hits.

Constructing a Confidence Interval

The greater the confidence, the wider the interval.

Think of a weather forecast: The high temperature tomorrow will between 40 below 0 and 200 above! That will have 100% confidence, but a lot of

variability!Although 80% has less variability, it is less

accurate. That’s why statisticians typically use 90% or above – most often 95%.

Constructing a Confidence Interval

Recall that the confidence interval is constructed using: estimate ± margin of error

Using a 95% confidence interval, we can rewrite that

± 2 •

A more general formula is:statistic ± (critical value) • (standard deviation of statistic)

x x

Homework Problems

Read textbook pages p. 469 – 478

Complete exercises:p. 481 – 483 #1, 2, 5, 7, 9, 11, 13, 15,

16

Check answers to the odd problems.