The population mean \(\mu\) (the greek letter "mu") and the population proportion p are two different population parameters. For example:

• We might be interested in learning about \(\mu\), the average weight of all middle-aged female Americans. The population consists of all middle-aged female Americans, and the parameter is µ.
• Or, we might be interested in learning about p, the proportion of likely American voters approving of the president's job performance. The population comprises all likely American voters, and the parameter is p.

The problem is that 99.999999999999... % of the time, we don't — or can't — know the real value of a population parameter. The best we can do is estimate the parameter! This is where samples and statistics come in to play.

Samples and statistics

The sample mean, \(\bar{x}\), and the sample proportion \(\hat{p}\) are two different sample statistics. For example:

• We might use \(\bar{x}\), the average weight of a random sample of 100 middle-aged female Americans, to estimate µ, the average weight of all middle-aged female Americans.
• Or, we might use \(\hat{p}\), the proportion in a random sample of 1000 likely American voters who approve of the president's job performance, to estimate p, the proportion of all likely American voters who approve of the president's job performance.

Because samples are manageable in size, we can determine the actual value of any statistic. We use the known value of the sample statistic to learn about the unknown value of the population parameter.

There are two ways to learn about a population parameter.

1) We can use confidence intervals to estimate parameters.

"We can be 95% confident that the proportion of Penn State students who have a tattoo is between 5.1% and 15.3%."

2) We can use hypothesis tests to test and ultimately draw conclusions about the value of a parameter.

"There is enough statistical evidence to conclude that the mean normal body temperature of adults is lower than 98.6 degrees F."

We review these two methods in the next two sections.

Let's review the basic concept of a confidence interval.

Suppose we want to estimate an actual population mean \(\mu\). As you know, we can only obtain \(\bar{x}\), the mean of a sample randomly selected from the population of interest. We can use \(\bar{x}\) to find a range of values:

\[\text{Lower value} < \text{population mean}\;\; \mu < \text{Upper value}\]

that we can be really confident contains the population mean \(\mu\). The range of values is called a "confidence interval."

The previous example illustrates the general form of most confidence intervals, namely:

$\text{Sample estimate} \pm \text{margin of error}$

The lower limit is obtained by:

$\text{the lower limit L of the interval} = \text{estimate} - \text{margin of error}$

The upper limit is obtained by:

$\text{the upper limit U of the interval} = \text{estimate} + \text{margin of error}$

Once we've obtained the interval, we can claim that we are really confident that the value of the population parameter is somewhere between the value of L and the value of U.

So far, we've been very general in our discussion of the calculation and interpretation of confidence intervals. To be more specific about their use, let's consider a specific interval, namely the "t-interval for a population mean µ."

(1-α)100% t-interval for the population mean \(\mu\)

If we are interested in estimating a population mean \(\mu\), it is very likely that we would use the t-interval for a population mean \(\mu\).

Note that:

• the "t-multiplier," which we denote as \(t_{\alpha/2, n-1}\), depends on the sample size through n - 1 (called the "degrees of freedom") and the confidence level \((1-\alpha)\times100%\) through \(\frac{\alpha}{2}\).
• the "standard error," which is \(\frac{s}{\sqrt{n}}\), quantifies how much the sample means \(\bar{x}\) vary from sample to sample. That is, the standard error is just another name for the estimated standard deviation of all the possible sample means.
• the quantity to the right of the ± sign, i.e., "t-multiplier × standard error," is just a more specific form of the margin of error. That is, the margin of error in estimating a population mean µ is calculated by multiplying the t-multiplier by the standard error of the sample mean.
• the formula is only appropriate if a certain assumption is met, namely that the data are normally distributed.

Clearly, the sample mean \(\bar{x}\), the sample standard deviation s, and the sample size n are all readily obtained from the sample data. Now, we need to review how to obtain the value of the t-multiplier, and we'll be all set.

How is the t-multiplier determined?

As the following graph illustrates, we put the confidence level $1-\alpha$ in the center of the t-distribution. Then, since the entire probability represented by the curve must equal 1, a probability of α must be shared equally among the two "tails" of the distribution. That is, the probability of the left tail is $\frac{\alpha}{2}$ and the probability of the right tail is $\frac{\alpha}{2}$. If we add up the probabilities of the various parts $(\frac{\alpha}{2} + 1-\alpha + \frac{\alpha}{2})$, we get 1. The t-multiplier, denoted \(t_{\alpha/2}\), is the t-value such that the probability "to the right of it" is $\frac{\alpha}{2}$:

It should be no surprise that we want to be as confident as possible when we estimate a population parameter. This is why confidence levels are typically very high. The most common confidence levels are 90%, 95%, and 99%. The following table contains a summary of the values of \(\frac{\alpha}{2}\) corresponding to these common confidence levels. (Note that the"confidence coefficient" is merely the confidence level reported as a proportion rather than as a percentage.)

Think about the width of the interval in the previous example. In general, do you think we desire narrow confidence intervals or wide confidence intervals? If you are not sure, consider the following two intervals:

• We are 95% confident that the average GPA of all college students is between 1.0 and 4.0.
• We are 95% confident that the average GPA of all college students is between 2.7 and 2.9.

Which of these two intervals is more informative? Of course, the narrower one gives us a better idea of the magnitude of the true unknown average GPA. In general, the narrower the confidence interval, the more information we have about the value of the population parameter. Therefore, we want all of our confidence intervals to be as narrow as possible. So, let's investigate what factors affect the width of the t-interval for the mean \(\mu\).

Of course, to find the width of the confidence interval, we just take the difference in the two limits:

Width = Upper Limit - Lower Limit

What factors affect the width of the confidence interval? We can examine this question by using the formula for the confidence interval and seeing what would happen should one of the elements of the formula be allowed to vary.

\[\bar{x}\pm t_{\alpha/2, n-1}\left(\dfrac{s}{\sqrt{n}}\right)\]

What is the width of the t-interval for the mean? If you subtract the lower limit from the upper limit, you get:

\[\text{Width }=2 \times t_{\alpha/2, n-1}\left(\dfrac{s}{\sqrt{n}}\right)\]

Now, let's investigate the factors that affect the length of this interval. Convince yourself that each of the following statements is accurate:

• As the sample mean increases, the length stays the same. That is, the sample mean plays no role in the width of the interval.
• As the sample standard deviation s decreases, the width of the interval decreases. Since s is an estimate of how much the data vary naturally, we have little control over s other than making sure that we make our measurements as carefully as possible.
• As we decrease the confidence level, the t-multiplier decreases, and hence the width of the interval decreases. In practice, we wouldn't want to set the confidence level below 90%.
• As we increase the sample size, the width of the interval decreases. This is the factor that we have the most flexibility in changing, the only limitation being our time and financial constraints.

In Closing

In our review of confidence intervals, we have focused on just one confidence interval. The important thing to recognize is that the topics discussed here — the general form of intervals, determination of t-multipliers, and factors affecting the width of an interval — generally extend to all of the confidence intervals we will encounter in this course.

In reviewing hypothesis tests, we start first with the general idea. Then, we keep returning to the basic procedures of hypothesis testing, each time adding a little more detail.

The general idea of hypothesis testing involves:

1. Making an initial assumption.
2. Collecting evidence (data).
3. Based on the available evidence (data), deciding whether to reject or not reject the initial assumption.

Every hypothesis test — regardless of the population parameter involved — requires the above three steps.

Did you notice the use of the phrase "behave as if" in the previous discussion? We "behave as if" the defendant is guilty; we do not "prove" that the defendant is guilty. And, we "behave as if" the defendant is innocent; we do not "prove" that the defendant is innocent.

This is a very important distinction! We make our decision based on evidence not on 100% guaranteed proof. Again:

• If we reject the null hypothesis, we do not prove that the alternative hypothesis is true.
• If we do not reject the null hypothesis, we do not prove that the null hypothesis is true.

We merely state that there is enough evidence to behave one way or the other. This is always true in statistics! Because of this, whatever the decision, there is always a chance that we made an error.

Let's review the two types of errors that can be made in criminal trials:

Table S.3.2 shows how this corresponds to the two types of errors in hypothesis testing.

Note that, in statistics, we call the two types of errors by two different  names -- one is called a "Type I error," and the other is called  a "Type II error." Here are the formal definitions of the two types of errors:

There is always a chance of making one of these errors. But, a good scientific study will minimize the chance of doing so!

Recall that it is either likely or unlikely that we would observe the evidence we did given our initial assumption. If it is likely, we do not reject the null hypothesis. If it is unlikely, then we reject the null hypothesis in favor of the alternative hypothesis. Effectively, then, making the decision reduces to determining "likely" or "unlikely."

In statistics, there are two ways to determine whether the evidence is likely or unlikely given the initial assumption:

1. We could take the "critical value approach" (favored in many of the older textbooks).
2. Or, we could take the "P-value approach" (what is used most often in research, journal articles, and statistical software).

In the next two sections, we review the procedures behind each of these two approaches. To make our review concrete, let's imagine that μ is the average grade point average of all American students who major in mathematics. We first review the critical value approach for conducting each of the following three hypothesis tests about the population mean $\mu$:

In Practice

• We would want to conduct the first hypothesis test if we were interested in concluding that the average grade point average of the group is more than 3.
• We would want to conduct the second hypothesis test if we were interested in concluding that the average grade point average of the group is less than 3.
• And, we would want to conduct the third hypothesis test if we were only interested in concluding that the average grade point average of the group differs from 3 (without caring whether it is more or less than 3).

Do you remember how to test the independence of two categorical variables? This test is performed by using a Chi-square test of independence.

Recall that we can summarize two categorical variables within a two-way table, also called an r × c contingency table, where r = number of rows, c = number of columns. Our question of interest is “Are the two variables independent?” This question is set up using the following hypothesis statements:

We will compare the value of the test statistic to the critical value of \(\chi_{\alpha}^2\) with the degree of freedom = (r - 1) (c - 1), and reject the null hypothesis if \(\chi^2 \gt \chi_{\alpha}^2\).

Consider a research experiment where the p-value computed from the data was 0.12. As a result, one would fail to reject the null hypothesis because this p-value is larger than \(\alpha\) = 0.05. However, there still exist two possible cases for which we failed to reject the null hypothesis:

1. the null hypothesis is a reasonable conclusion,
2. the sample size is not large enough to either accept or reject the null hypothesis, i.e., additional samples might provide additional evidence.

Power analysis is the procedure that researchers can use to determine if the test contains enough power to make a reasonable conclusion. From another perspective power analysis can also be used to calculate the number of samples required to achieve a specified level of power.

If the sample size is fixed, then decreasing Type I error \(\alpha\) will increase Type II error \(\beta\). If one wants both to decrease, then one has to increase the sample size.

To calculate the smallest sample size needed for specified \(\alpha\), \(\beta\), \(\mu_a\), then (\(\mu_a\) is the likely value of \(\mu\) at which you want to evaluate the power.

Let's investigate by returning to our previous example.

In summary, you should see how power analysis is very important so that we are able to make the correct decision when the data indicate that one cannot reject the null hypothesis. You should also see how power analysis can also be used to calculate the minimum sample size required to detect a difference that meets the needs of your research.

Let us consider the parameter p of the population proportion. For instance, we might want to know the proportion of males within a total population of adults when we conduct a survey. A test of proportion will assess whether or not a sample from a population represents the true proportion of the entire population.

The steps to perform a test of proportion using the critical value approval are as follows:

1. State the null hypothesis H₀ and the alternative hypothesis H_A.
2. Calculate the test statistic:

   \[z=\frac{\hat{p}-p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}\]

   where \(p_0\) is the null hypothesized proportion i.e., when \(H_0: p=p_0\)

3. Determine the critical region.

4. Make a decision. Determine if the test statistic falls in the critical region. If it does, reject the null hypothesis. If it does not, do not reject the null hypothesis.

Next, let's state the procedure in terms of performing a proportion test using the p-value approach. The basic procedure is:

1. State the null hypothesis H₀ and the alternative hypothesis H_A.
2. Set the level of significance \(\alpha\).
3. Calculate the test statistic:

   \[z=\frac{\hat{p}-p_o}{\sqrt{\frac{p_0(1-p_0)}{n}}}\]

4. Calculate the p-value.

5. Make a decision. Check whether to reject the null hypothesis by comparing the p-value to \(\alpha\). If the p-value < \(\alpha\) then reject \(H_0\); otherwise do not reject \(H_0\).

We suggest you...

1. Review the concepts and methods on the pages in this section of this website.
2. Download and Complete the Self-Assessment Exam.
3. Determine your Score by reviewing the Self-Assessment Exam Solutions

A score below 70% suggests that the concepts and procedures that are covered in STAT 500 have not been mastered adequately. Students are strongly encouraged to take STAT 500, thoroughly review the materials that are covered in the sections above or take additional coursework that focuses on these foundations.

If you have struggled with the concepts and methods that are presented here, you will indeed struggle in any of the graduate-level courses included in the Master of Applied Statistics program above STAT 500 that expect and build on this foundation.