AP Statistics – Ch 8.3 Notes Name: ___________________ Estimating a Population Mean

When σ Is Unknown: The t Distributions When we standardize based on the sample standard deviation sx, our statistic has a new distribution called a t distribution.

ü It has a different shape than the standard Normal curve: ü It is symmetric with a single peak at 0, ü However, it has much more area in the tails.

Like any standardized statistic, t tells us how far !̅is from its mean μ in standard deviation units. There is a different distribution for each sample size, specified by its degrees of freedom (df) The t Distributions; Degrees of Freedom When we perform inference about a population mean µ using a t distribution, the appropriate degrees of freedom are found by subtracting 1 from the sample size n, making df = n - 1. We will write the t distribution with n - 1 degrees of freedom as tn-1. Conditions for Constructing a Confidence Interval about a Proportion

Practice Problem: What critical value t* from table B should be used in constructing a confidence interval for the population mean in each of the following settings? 1. A 95% confidence interval based on an SRS of size n=12?

2. A 90% confidence interval from a random sample of 48 observations.

Draw an SRS of size n from a large population that has a Normal distribution with mean µ and standard deviation σ. The statistic

has the t distribution with degrees of freedom df = n – 1. When the population distribution isn’t Normal, this statistic will have approximately a t

n – 1

distribution if the sample size is large enough.

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t =x −µ

sx n

ü The density curves of the t distributions are similar in shape to the standard Normal curve.

ü The spread of the t distributions is a bit greater than that of the standard Normal distribution.

ü The t distributions have more probability in the tails and less in the center than does the standard Normal.

ü As the degrees of freedom increase, the t density curve approaches the standard Normal curve ever more closely.

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AP Statistics – Ch 8.3 Notes Name: ___________________ Estimating a Population Mean

Conditions for estimating μ

Practice Problem: Determine whether we can safely use a t* critical value to calculate a confidence interval for the population mean in each of the following settings. 1. We want to estimate the average time (in minutes) to order and receive a regular coffee at a local

coffee shop. The time for five randomly selected visits to a local coffee shop are shown in the dotplot below.

2. We want to estimate the average height of high school students. The boxplot below shows the

distribution of heights (in cm) for 100 randomly selected students.

3. A middle-school counselor wants to estimate how many minutes, on average, students spend doing homework. A histogram of the amount of time doing homework the previous evening for a random sample of 20 students is shown below.

One-Sample t Interval for a Population Mean

• Random: The data come from a well-designed random sample or randomized experiment. o 10%: When sampling without replacement, check that

• Normal/Large Sample: The population has a Normal distribution or the sample size is large (n ≥ 30). If the population distribution has unknown shape and n < 30, use a graph of the sample data to assess the Normality of the population. Do not use t procedures if the graph shows strong skewness or outliers.

n ≤ 110

N

When the conditions are met, a C% confidence interval for the unknown mean µ is

where t* is the critical value for the tn-1

distribution with C% of its area between −t* and t*.

x ± t * sxn

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AP Statistics – Ch 8.3 Notes Name: ___________________ Estimating a Population Mean

Practice Problem: Environmentalists, government officials, and vehicle manufacturers are all interested in studying the auto exhaust emissions produced by motor vehicles. The major pollutants in auto exhaust from gasoline engines are hydrocarbons, carbon monoxide, and nitrogen oxides (NOX). Researchers collected data on the NOX levels (in grams/mile) for a random sample of 40 light-duty engines of the same type. The mean NOX reading was 1.2675 and the standard deviation was 0.3332. Problem: Construct and interpret a 95% confidence interval for the mean amount of NOX emitted by light-duty engines of this type.

Choosing Sample Size for a Desired 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 time-consuming and expensive, so the researchers want to know the minimum number of monkeys they will need to generate a satisfactory estimate.

To determine the sample size n that will yield a level C confidence interval for a population mean with a specified margin of error ME: • Get a reasonable value for the population standard deviation σ from an earlier or pilot study. • Find the critical value z* from a standard Normal curve for confidence level C. • Set the expression for the margin of error to be less than or equal to ME and solve for n:

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z * σn≤ ME

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