Point Estimate

A single value that represents some unknown characteristic, such as the population mean.

A random sample of 200 graduates of U.S. colleges reveals a mean annual income of $62,600. What is the best estimate of the unknown mean annual income for all graduates?

$62,600

Confidence Interval (CI)

A range of values that, with a known degree of certainty, includes an unknown population characteristic, such as a population mean.

Reading achievement scores are obtained for a group of fourth graders. A score of 4.0 indicates a level of achievement appropriate for fourth grade, a score below 4.0 indicates underachievement, and a score above 4.0 indicates overachievement. Assume that the population standard deviation equals 0.4.A random sample of 64 fourth graders reveals a mean achievement score of 3.82.

a) Construct a 95 percent confidence interval for the unknown population mean. (Remeber to convert the standard deviation to a standard error.)

b) Interpret this confidence interval: That is, do you find consistent evidence either of overachievement or of underachievement?

a) 3.82 ± 1.96(.4/√64) = 3.92 and 3.72

b) We can claim, with 95 percent confidence, that the interval between 3.72 and 3.92 includes the true population mean reading score for the fourth graders. All of these values suggest that, on average, the fourth graders are underachieving.

Before taking the GRE, a random sample of college seniors received special training on how to take the test. After analyzing their scores on the GRE, the investigator reported a dramatic gain, relative to the national average of 500, as indicated by a 95 percent confidence interval of 507 to 527. Is the following interpretation true or false?

About 95 percent of all subjects scored between 507 and 527.

False. We can be 95 percent confident that the mean for all subjects will be between 507 and 527.

Before taking the GRE, a random sample of college seniors received special training on how to take the test. After analyzing their scores on the GRE, the investigator reported a dramatic gain, relative to the national average of 500, as indicated by a 95 percent confidence interval of 507 to 527. Is the following interpretation true or false?

The interval from 507 to 527 refers to possible values of the population mean for all students who undergo special training.

True.

Before taking the GRE, a random sample of college seniors received special training on how to take the test. After analyzing their scores on the GRE, the investigator reported a dramatic gain, relative to the national average of 500, as indicated by a 95 percent confidence interval of 507 to 527. Is the following interpretation true or false?

The true population mean definitely is between 507 and 527.

False. We can be reasonably confident - but not absolutely confident - that the true population mean lies beteen 507 and 527.

This particular interval describes the population mean about 95 percent of the time.

False. This particular interval either describes the one true population mean or fails to describe the one true population mean.

In practice, we never really know whether the interval from 507 to 527 is true or false.

True.

We can be reasonably confident that the population mean is between 507 and 527.

True.

Level of Confidence

The percent of time that a series of confidence intervals includes the unknown population characteristic, such as the population mean.

On the basis of a random sample of 120 adults, a pollster reports, with 95 percent confidence, that between 58 and 72 percent of all Americans belive in life after death.

If this interval is too wide, what, if anything, can be done with the existing data to obtain a narrower confidence interval?

Switch to an interval having a lesser degree of confidence, such as 90 percent or 75 percent.

On the basis of a random sample of 120 adults, a pollster reports, with 95 percent confidence, that between 58 and 72 percent of all Americans belive in life after death.

What can be done to obtain a narrower 95 percent confidence interval if another similar investigation is being planned?

Increase the sample size.

Margin of Error

That which is added to and subtracted from some sample value, such as the sample proportion or sample mean, to obtain the limits of a confidence interval.

In a recent scientific sample of about 900 adult Americans, 70 percent favor stricter fun control of assault weapons, with a margin of error of ± 4 percent for a 95 percent confidence interval. Therefore, the 95 percent confidence interval equals 66 to 74 percent. Indicate whether the following interpretation is true or false:

The interval from 66 to 74 percent refers to possible values of the sample percent.

False. The interval from 66 to 74 percent refers to possible values of the population proportion.

In a recent scientific sample of about 900 adult Americans, 70 percent favor stricter fun control of assault weapons, with a margin of error of ± 4 percent for a 95 percent confidence interval. Therefore, the 95 percent confidence interval equals 66 to 74 percent. Indicate whether the following interpretation is true or false:

The true population percent is between 66 and 74 percent.

False. We can be reasonably confident - but not absolutely confident- that the true population proportion is between 66 and 74 percent.

In a recent scientific sample of about 900 adult Americans, 70 percent favor stricter fun control of assault weapons, with a margin of error of ± 4 percent for a 95 percent confidence interval. Therefore, the 95 percent confidence interval equals 66 to 74 percent. Indicate whether the following interpretation is true or false:

In the long run, a series of intervals similar to this one would fail to include the population percent about 5 percent of the time.

True.

We can be reasonably confident that the population percent is between 66 and 74 percent.

True.