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Estimates

In the previous sections we dealt with sample statistics. For instance the mean and standard deviation of those samples were determined. In this section we discuss how a sample statistic can be used to estimate a population parameter. We also determine how good that the estimate is. There is no reasonable way to get the exact population parameter.

Confidence Intervals

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Suppose 100 different samples were taken from the same population. Do you think the means for each sample will be the same? In most cases the answer is no. Which of the 100 means of the samples should be used as the estimate for the population mean? The answer is any sample mean. How can that be? This is one of the most interesting (and confusing) parts of statistics. There is no way to guarantee that the sample that we choose is the best one. But there is a CLEVER way to determine intervals (one for each sample), so that the population mean, $\mu$, will be contained in approximately 95% (or 90% or 99%, which ever we choose) of those intervals. The 95% is called the degree of confidence. The greek letter alpha, $\alpha$, is used to represent a probability or area. The value of $\alpha$ is the compliment of the dgree of confidence.

A Basic Rule

A sample mean, x, is the best estimate for the population mean, $\mu$.

Definitions

An estimator is a formula or process for using sample data to estimate a population parameter.

An estimate is a specific value (or a range of values) used to approximate a population parameter.

A confidence interval is a range of values used to estimate the true value of a population parameter.

The degree of confidence is the probability (1-$\alpha$) that is described above.

The Critical Value

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A critical value is a Z-score that is associated with $\alpha$. The Z-score that is needed is denoted by $z_{\alpha /2}$.

Definition

A critical value, $z_{\alpha /2}$, is a Z-score that has the property that the area between $-z_{\alpha /2}$ and  $z_{\alpha /2}$ is 1-$\alpha$. The reason for the $\alpha$/2 is that the area of each of the remaining regions (the left and right tails) both have area $\alpha$/2.

Some common critical values are 1.645, 1.96 and 2.575. These values correspond to 90%, 95% and 99% degree of confidence, respectively. Check it out for yourself.

The Error

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Margin of Error

The margin of Error E is found by using the standard deviation, $\sigma$, the sample size n, the degree of confidence, and the critical value. The exact formula is:

$$E=z_{\alpha /2}\cdot\dfrac{\sigma}{\sqrt{n}}$$ (5.1)

How to Calculate an Error

Suppose 150 college students were asked their Intelligence Quotient (IQ). From the data you find out that $\sigma$=11.3. What are the levels of error associated with a 95% confidence interval? Solution: Using formula (5.1) with n=150, $z_{\alpha /2}$=1.96, and $\sigma$=11.3, the error is 1.81.

What does the error mean?

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The idea of this section was to estimate the mean of the population from the mean of a particular sample. In the example above, the sample mean, x can be calculated. The interval in question is then calculated from $\mu$ and E.

How to Calculate a Confidence Interval

The end points of the confidence interval is found by using x and E. The interval is

( x - E, x + E)

A confidence interval

Using the information from above, what is the 95% confidence interval? Solution: There is not enough information above (the sample mean is missing). From the data, the sample mean can be computed. IF the sample mean turns out to be 120, then the confidence interval is (118.19, 121.81).

How to Interpret a Confidence Interval

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The interpretation of a confidence interval seems to be the most confusing part of confidence intervals. Most students (at least initially) have the wrong interpretation. The most popular WRONG way to interpret a confidence interval is that there is a 95% chance that the population mean is in the interval.

The interpretation of a confidence interval

Interpret the confidence interval found in the IQ example Solution: The confidence interval for the IQ scores is (118.19, 121.81).

The correct way:	If many different samples of size 150 were taken and confidence intervals were constructed for each of those samples, then 95% of those samples will contain the population mean. Thus, we say "we are 95% confident that the interval (118.19, 121.81) actually contains the population mean."
Wrong way:	There is a 95% chance that the population mean is in the interval.

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