Sampling distributions rule of thumb…

Sampling distributions rule of thumb…

Some important points about sample distributions…

If we obtain a sample that meets the rules of thumb, then…

In statistics, estimation refers to the process by which one makes inferences about a population, based on information obtained from a sample.

Point Estimate vs. Interval Estimate

Statisticians use sample statistics to estimate population parameters. For example, sample means are used to estimate population means; sample proportions, to estimate population proportions.

An estimate of a population parameter may be expressed in two ways:

•Point estimate. A point estimate of a population parameter is a single value of a statistic. For example, the sample mean x is a point estimate of the population mean μ. Similarly, the sample proportion p is a point estimate of the population proportion P.

•Interval estimate. An interval estimate is defined by two numbers, between which a population parameter is said to lie. For example, a < x < b is an interval estimate of the population mean μ. It indicates that the population mean is greater than a but less than b.

Suppose you take a SRS of 100 students from Roy High and calculate their mean GPA to 2.75. What can you say about the mean GPA of all students (μ) at Roy High?

Confidence Intervals

Statisticians use a confidence interval to express the precision and uncertainty associated with a particular sampling method. A confidence interval consists of three parts.

• A confidence level. • A statistic. • A margin of error.

The confidence level describes the uncertainty of a sampling method. The statistic and the margin of error define an interval estimate that describes the precision of the method. The interval estimate of a confidence interval is defined by the sample statistic + margin of error.

For example, we might say that we are 95% confident that the true population mean falls within a specified range. This statement is a confidence interval. It means that if we used the same sampling method to select different samples and compute different interval estimates, the true population mean would fall within a range defined by the sample statistic + margin of error 95% of the time.

Confidence intervals are preferred to point estimates, because confidence intervals indicate (a) the precision of the estimate and (b) the uncertainty of the estimate.

Let’s take our hypothetical SRS of 100 students from Roy High with mean GPA to 2.75. Let’s continue on by saying that we happen to know that the standard deviation of Roy High students’ GPA is 0.54. What can we say about the distribution of Roy High students’ GPA?

Confidence Level

The probability part of a confidence interval is called a confidence level. The confidence level describes how strongly we believe that a particular sampling method will produce a confidence interval that includes the true population parameter.

Here is how to interpret a confidence level. Suppose we collected many different samples, and computed confidence intervals for each sample. Some confidence intervals would include the true population parameter; others would not. A 95% confidence level means that 95% of the intervals (samples) contain the true population parameter; a 90% confidence level means that 90% of the intervals (samples) contain the population parameter; and so on.

Check out the diagram on pg. 541.

Margin of Error

In a confidence interval, the range of values above and below the sample statistic is called the margin of error.

Thus, the format for every confidence interval is this:

estimate ± margin of error

To find the confidence interval, you must be able to find the standard error. This is the first step in calculating a confidence interval.

Notation

The following notation is helpful, when we talk about the standard deviation and the standard error.

Standard Deviation of Sample Estimates

Statisticians use sample statistics to estimate population parameters. Naturally, the value of a statistic may vary from one sample to the next.The variability of a statistic is measured by its standard deviation. The table below shows formulas for computing the standard deviation of statistics from simple random samples. These formulas are valid when the population size is much larger (at least 10 times larger) than the sample size.

Let’s take our hypothetical SRS of 100 students from Roy High with mean GPA to 2.75. Let’s continue on by saying that we happen to know that the standard deviation of Roy High students’ GPA is 0.54. What is the standard error of this sample?

In a confidence interval, the range of values above and below the sample statistic is called the margin of error.

For example, suppose we wanted to know the percentage of adults that exercise daily. We could devise a sample design to ensure that our sample estimate will not differ from the true population value by more than, say, 5 percent (the margin of error) 90 percent of the time (the confidence level).

How to Compute the Margin of Error

The margin of error can be defined by either of the following equations.

Margin of error = Critical value x Standard deviation

How to Find the Critical Value

The critical value is a factor used to compute the margin of error. This section describes how to find the critical value, when the sampling distribution of the statistic is normal or nearly normal.The central limit theorem states that the sampling distribution of a statistic will be normal or nearly normal, if any of the following conditions apply.

•The population distribution is normal. •The sampling distribution is symmetric, unimodal, without outliers, and the sample size is 15 or less. •The sampling distribution is moderately skewed, unimodal, without outliers, and the sample size is between 16 and 40. •The sample size is greater than 40, without outliers.

SOME USEFUL CRITICAL VALUES TO MEMORIZE!

Confidence Level

Tail Area

z*

90% 0.05 1.645

95% 0.025 1.960

99% 0.005 2.576

Let’s take our hypothetical SRS of 100 students from Roy High with mean GPA to 2.75. Let’s continue on by saying that we happen to know that the standard deviation of Roy High students’ GPA is 0.54. Are the conditions met for which we can find the margin of error?

Statisticians use a confidence interval to describe the amount of uncertainty associated with a sample estimate of a population parameter.

How to Interpret Confidence Intervals

Consider the following confidence interval: We are 90% confident that the population mean is greater than 100 and less than 200.

Some people think this means there is a 90% chance that the population mean falls between 100 and 200. This is incorrect. Like any population parameter, the population mean is a constant, not a random variable. It does not change.

The probability that a constant falls within any given range is always 0.00 or 1.00.The confidence level describes the uncertainty associated with a sampling method. Suppose we used the same sampling method to select different samples and to compute a different interval estimate for each sample. Some interval estimates would include the true population parameter and some would not. A 90% confidence level means that we would expect 90% of the interval estimates to include the population parameter; A 95% confidence level means that 95% of the intervals would include the parameter; and so on.

Confidence Interval Data Requirements

To express a confidence interval, you need three pieces of information.

•Confidence level •Statistic •Margin of error

Given these inputs, the range of the confidence interval is defined by the sample statistic + margin of error. And the uncertainty associated with the confidence interval is specified by the confidence level.Most often, the margin of error is not given; you must calculate it.

How to Construct a Confidence Interval

There are four steps to constructing a confidence interval. Identify a sample statistic. • Choose the statistic (e.g, mean, standard deviation) that you will use to estimate a population parameter.

• Select a confidence level. As we noted in the previous section, the confidence level describes the uncertainty of a sampling method. Often, researchers choose 90%, 95%, or 99% confidence levels; but any percentage can be used.

• Find the margin of error. If you are working on a homework problem or a test question, the margin of error may be given. Often, however, you will need to compute the margin of error, based on one of the following equations.

Margin of error = Critical value * Standard deviation of statistic Margin of error = Critical value * Standard error of statistic

• Specify the confidence interval. The uncertainty is denoted by the confidence level. And the range of the confidence interval is defined by the following equation.

Confidence interval = sample statistic + Margin of error

Let’s take our hypothetical SRS of 100 students from Roy High with mean GPA to 2.75. Let’s continue on by saying that we happen to know that the standard deviation of Roy High students’ GPA is 0.54. Find a 95% confidence interval and interpret it in context.