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The Practice of Statistics, 5th Edition Starnes, Tabor, Yates, Moore

Bedford Freeman Worth Publishers

CHAPTER 8Estimating with Confidence8.3Estimating a Population Mean

Learning Objectives

After this section, you should be able to:

The Practice of Statistics, 5 th

Edition2

$ STATE and CHECK the Random, 10%, and Normal/Large Sample conditions for constructing a confidence interval for a population mean.

$ EXPLAIN how the t distributions are different from the standard Normal distribution and why it is necessary to use a t distribution when calculating a confidence interval for a population mean.

$ DETERMINE critical values for calculating a C% confidence interval for a population mean using a table or technology.

$ CONSTRUCT and INTERPRET a confidence interval for a population mean.

$ DETERMINE the sample size required to obtain a C% confidence interval for a population mean with a specified margin of error.

Estimating a Population Mean

Feb 23­2:24 PM

Proportion Mean

Sampling distribution of the sample

When p is unknown, use as an estimate to calculate σ.

When σ is unknown, use sx as an estimate.

Feb 23­2:43 PM

The problem with Z:

We need to know the population mean and standard deviation. In smaller samples, the margins of error get messed up by using a z calculated with the sample mean and SD.

Enter t:

Sort of an estimated Z. Gives better results,especially when the sample size is small.

Feb 25­10:52 AM

• Based on probability theory• Two steps

1. Take a given population and draw random samples again and again

2. Plot the means from the results of Step 1 and it will be a normal curve where the center of the curve is the mean and the variation represents the standard error

• Even if the population distribution is skewed, the distribution from Step 2 will be normal!

Central Limit Theorem

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Feb 25­10:47 AM

What is t?

• “Cousin” of the z statistic that does not require the population mean (μ) or variance (σ2)to be known

• Can be used to calculate a confidence interval for a population (when the only information about the population comes from the sample)

Feb 25­10:50 AM

• For the t statistic,> We use sample data to compute an “Estimated

Standard Error of the Mean”> Uses the exact same formula but substitutes the

sample variance for the unknown population variance and/or standard deviation

Feb 25­12:18 PM

• Think of t as an estimated z score• Estimation is due to the unknown standard deviation• With large samples, the estimation is good and the t statistic is very close to z

• In smaller samples, the estimation is poorer• Why?• Degrees of freedom is used to describe how well t represents z

Feb 25­12:21 PM

Feb 25­12:21 PM

• df = n – 1• Value of df will determine how well the distribution of t approximates a normal one> With larger df’s, the distribution of the t statistic will

approximate the normal curve> With smaller df’s, the distribution of t will be flatter and

more spread out• t table uses critical values and incorporates df

Degrees of FreedomWhen σ Is Unknown: The t Distributions

.Slightly different shape than the standard Normal curve:

It is symmetric with a single peak at 0,However, it has much more area in the tails.

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The Practice of Statistics, 5 th

Edition6The t Distributions; Degrees of Freedom

t

The smaller the sample size the more we must stretch out the tails!

Example: Using the calculator to find Critical t* ValuesProblem: What critical value t* should be used in constructing a confidence interval for the population meanfor a 95% confidence interval based on an SRS of size n = 12?

Solution: Use InvT (2nd VARS). The area in each tail will be (1 ­ 0.95)/2 = 0.025.

Degrees of freedom will be 12 ­ 1 = 11.

t* = ­2.201

Feb 25­12:43 PM

What critical value t* should be used to construct a 90% confidence interval from a random sample of 48 observations?

­1.678

Feb 25­1:12 PM

Check your conditions:

Random ­ does the data come from a random sample or experiment

10% ­ when sampling without replacement, n ≤ 0.1 N

Normal/Large Sample:

n ≥ 30 OR

assess normality by looking at a graph. If strongly skewed or outliers, do no use t procedures.

Mar 9­12:06 PM

To assess normality for small samples (n < 30), make a graph. What type of graph? It doesn't matter ­ dot plot, histogram, etc. Look for outliers and strong skewness.

Mar 16­1:50 PM

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Feb 25­1:17 PM Feb 25­1:38 PM

Note: On the AP Exam, you will NOT be asked to construct a confidence interval when the conditions are not met. If you have concerns about an interval, state them but construct the interval anyway. In many cases, students get mixed up about the conditions. Don't lose points for NOT constructing the interval.

Feb 25­1:18 PM

Example:

As part of their final project in AP® Statistics, Christina and Rachel randomly selected 18 rolls of a generic brand of toilet paper to measure how well this brand could absorb water. To do this, they poured 1/4 cup of water onto a hard surface and counted how many squares of toilet paper it took to completely absorb the water. Here are the results from their 18 rolls:

29 20 25 29 21 24 27 25 2429 24 27 28 21 25 26 22 23

Construct and interpret a 99% confidence interval for the true mean number of squares of generic toilet paper needed to absorb 1/4 cup of water.

Feb 25­1:53 PM

On a free­response question, you are expected to use the 4­step process including defining the parameter, identifying the procedure and checking conditions.

Feb 25­1:21 PM

STATE: We want to estimate the true mean number of squares of generic toilet paper needed to absorb 1/4 cup of water with 99% confidence.

PLAN: We will construct a one­sample t interval if the conditions are met:

Random: The students selected the rolls of generic toilet paper at random.

10%: It is safe to assume that there are at least 10(18) = 180 rolls of generic toilet paper.Normal/Large Sample: Because the sample size is small (n = 18),

we need to check whether it is reasonable to believe that the population has a Normal distribution. The dotplot below doesn’t show any outliers or strong skewness, so it is reasonable to assume that the population distribution is approximately Normal

Feb 25­1:23 PM

Do: x = 24.94. sx = 2.86. df = 18 ­ 1 = 17. For 99% confidence, we will use t* = 2.898.

Conclude: We are 99% confident that the interval from 22.99 to 26.89 squares captures the true mean number of square of generic toilet paper needed to absorb 1/4 cup of water.

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Mar 9­11:45 AM

Even easier:

On the calculator: TInterval

Use either Data or Stats

Feb 25­1:50 PM

AP Common Errors

Students use technology to calculate an interval, and then try to show their work using an incorrect formula. If you include a formula, make sure that the results match.

Also, don't use the symbols unless you are certain about them. Just start with the numbers and solve.

Feb 25­1:41 PM

Choosing the Sample Size

Get a reasonable estimate of population SD from an earlier study

Find the critical value z from a standard Normal curve with confidence level

Set up your inequality and solve for n.

The Practice of Statistics, 5 th

Example: Determining sample size from margin of error

Researchers would like to estimate the mean cholesterol level µ of a particular variety of monkey that is often used in laboratory experiments.

They would like their estimate to be within 1 milligram per deciliter (mg/dl) of the true value of µ at a 95% confidence level.

A previous study involving this variety of monkey suggests that the standard deviation of cholesterol level is about 5 mg/dl.

Problem : Obtaining monkeys is time­consuming and expensive, so the researchers want to know the minimum number of monkeys they will need to generate a satisfactory estimate.

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Example: Determining sample size from margin of errorSolution: For 95% confidence, z* = 1.96.

We will use σ = 5 as our best guess for the standard deviation of the monkeys’ cholesterol level.

Set the expression for the margin of error to be at most 1 and solve for n :

Because 96 monkeys would give a slightly larger margin of error than desired, the researchers would need 97 monkeys to estimate the cholesterol levels to their satisfaction.

Section SummaryIn this section, we learned how to…

The Practice of Statistics, 5 th

Edition19

$ STATE and CHECK the Random, 10%, and Normal/Large Sample conditions for constructing a confidence interval for a population mean.

$ EXPLAIN how the t distributions are different from the standard Normal distribution and why it is necessary to use a t distribution when calculating a confidence interval for a population mean.

$ DETERMINE critical values for calculating a C% confidence interval for a population mean using a table or technology.

$ CONSTRUCT and INTERPRET a confidence interval for a population mean.

$ DETERMINE the sample size required to obtain a C% confidence interval for a population mean with a specified margin of error.

Estimating a Population Mean

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Mar 9­11:50 AM