7.1 ESTIMATING μ WHEN σ IS KNOWN

Chapter 7: Estimation

Estimating μ when σ is known Recall that we usually do not have access

to all measurements of an entire population, so we often rely on information from a sample.

Recall that estimation is a type of inference where we estimate the of a population parameter (6.4 page 292)

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Estimating μ when σ is known We use sample data to estimate

population parameters. In order to do this, there are some basic

assumptions that must be made

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Assumptions about the random variable

1. We have a simple random sample of size drawn from a population of values.

2. The value of σ, the population standard deviation of , is known.

3. If the distribution is normal, then our methods work for any sample size .

4. If has an unknown distribution, then we require a sample size . However, if the distribution is distinctly skewed and definitely not mound-shaped, a sample of size 50 or even 100 or higher may be necessary.

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Definitions A point estimate of a population parameter

is a single number estimate of the parameter. is the point estimate for μ

The value of a point estimate is usually not exactly equal to the population parameter.

The margin of error is the magnitude of the difference between the sample point estimate and the true population parameter. The margin of error for the mean is

If the population parameter is unknown, then we cannot calculate an exact margin of error

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Definitions The reliability of an estimate will be

measured by a confidence level, . Common confidence levels are: , , and

The critical value, , is the number such that the area under the standard normal curve between and equals . The area under the normal curve

between –zc and zc is the

probability that the standardized normal variable z lies in that interval:

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Figure 7.1

How to Find a Critical Value

1. Sketch a picture

2. Find the area in the left tail:

3. Evaluate invNorm() to get

Not in Textbook!

Note: Table 7-2 on page 336 is a useful source, it is also printed with Table 5

Example 1 – Find a Critical Value

Find a number z0.99 such that 99% of the area under the standard normal curve lies between –z0.99 and z0.99. That is, we will find z0.99 such that

P(–z0.99 < z < z0.99) = 0.99

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Solution:

Area Between –z and z Is 0.99

Figure 7-2

invNorm(.005) = –2.5758 ≈ –2.58

P(–z0.99 < z < z0.99) = 0.99

P(–2.58 < z < 2.58) = 0.99

Maximal Margin of Error In most problems μ is unknown (which is why

we are estimating it!) and so the margin of error is also unknown.

Using a level of confidence, we can say that the point estimate differs from the population mean by a maximal margin of error, .

Note: This formula assumes that the sampling distribution is exactly normal. If the distribution is not normal, then according to the central limit theorem, large samples () produce a distribution that is approximately normal.

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Confidence Intervals (in general)

A confidence interval is an interval estimate of a population parameter and is used to indicate the reliability of an estimate. We may get a different confidence interval for each

different sample that is taken. Once we have a confidence interval, it either does or does

not contain the population parameter. So, the probability is 1 or 0 that the interval contains the population parameter.

Once we have a confidence interval we say:“We are confident that we have one of the intervals that actually contains the population parameter”OR“At the confidence level, our interval contains the population parameter”

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Confidence Interval for μ when σ is Known

RequirementsLet x be a random variable appropriate to your application. Obtain

a simple random sample (of size n) of x values from which you compute the sample mean . The value of σ is already known (perhaps from a previous study).If you can assume that x has a normal distribution, then any sample size n will work. If you cannot assume this, then use a sample size of n ≥ 30.

Confidence Interval for μ when σ is known:

Where sample mean of a simple random sample

Confidence level

Critical Value for confidence level c

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How To Construct a Confidence Interval1. Check Requirements

Simple random sample? Assumption of normality? Sample size? Sample mean? Population standard deviation σ?

2. Compute 3. Construct the interval using

Not in Textbook!

Example 2 – Confidence Interval

Julia enjoys jogging. She has been jogging over a period of several years, during which time her physical condition has remained constantly good. Usually, she jogs 2 miles per day. The standard deviation of her times is = 1.80 minutes. During the past year, Julia has recorded her times to run 2 miles. She has a random sample of 90 of these times. For these 90 times, the mean was x = 15.60 minutes. Let be the mean jogging time for the entire distribution of Julia’s 2-mile running times (taken over the past year). Find a 0.95 confidence interval for .

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Solution – Confidence Interval

Simple random sample = 90 approximately normal σ = 1.80 = 15.60

We conclude with 95% confidence that the interval from 15.23 minutes to 15.97 minutes is one that contains the population mean of jogging times for Julia.

Using the Calculator1. Hit STAT, tab over TESTS,

Choose 7:Zinterval

2. Highlight STATS, hit ENTER

3. Enter the requested information

4. Highlight Calculate, Hit Enter

Note: The solution will be listed in the format (lower value, upper value)

Sample Size for Estimating the Mean μ In the design stages of statistical research

projects, it is a good idea to decide in advance on the confidence level you wish to use and to select the maximal margin of error E you want for your project. How you choose to make these decisions depends

on the requirements of the project and the practical nature of the problem.

Whatever specifications you make, the next step is to determine the sample size. Solving the maximal margin of error E formula for

enables us to determine the minimal sample size.

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How to Find the Sample Size n for Estimating μ when is σ known

Requirements The distribution is approximately normal

Formula for Sample Size

If is not a whole number, increase to the next higher whole number.

Note: is the minimal sample size for a specified confidence level and maximal error of estimate.

If n ≥ 30, then it is usually safe to use instead of σ

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Example 3 – Sample Size

A wildlife study is designed to find the mean weight of salmon caught by an Alaskan fishing company. A preliminary study of a random sample of 50 salmon showed s 2.15 pounds. How large a sample should be taken to be 99% confident that the sample mean is within 0.20 pound of the true mean weight ?

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Solution: A sample size of 770 will be large enough to satisfy the specifications. Of course, a sample size larger than 770 also works.

Assignment

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