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View the Reading on Confidence Intervals – Introduction (≈ 2575 words)

View the Reading on Comments about the Sample Mean (≈ 1500 words)

From the online version of Little Handbook of Statistical Practice.

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As we wrap up this topic, we wanted to again discuss the interpretation of a confidence interval.

What do we mean by “confidence”?

Suppose we find a 95% confidence interval for an unknown parameter, what does the 95% mean exactly?

- If we repeat the process for all possible samples of this size for the population, 95% of the intervals we construct will contain the parameter

This is NOT the same as saying “*the probability that μ (mu) is contained in (the interval constructed from my sample) is 95%.*” Why?!

**Answer:**

- Once we have a particular confidence interval, the true value is either in the interval constructed from our sample (probability = 1) or it is not (probability = 0). We simply do not know which it is. If we were to say “the probability that μ (mu) is contained in (the interval constructed from my sample) is 95%,” we know we would be incorrect since it is either 0 (No) or 1 (Yes) for any given sample. The probability comes from the “long run” view of the process.

- The probability we used to construct the confidence interval was based upon the fact that the sample statistic (x-bar, p-hat) will vary in a manner we understand (because we know the sampling distribution).

- The probability is associated with the randomness of our statistic so that for a particular interval we only speak of being “95% confident” which translates into an understanding about the process.

**In other words, in statistics, “95% confident” means our confidence in the process and implies that in the long run, we will be correct by using this process 95% of the time but that 5% of the time we will be incorrect. For one particular use of this process we cannot know if we are one of the 95% which are correct or one of the 5% which are incorrect. That is the statistical definition of confidence.**

- We can say that in the long run, 95% of these intervals will contain the true parameter and 5% will not.

**Example: **Suppose a 95% confidence interval for the proportion of U.S. adults who are not active at all is (0.23, 0.27).

**Correct Interpretation #1:**We are 95% confident that the true proportion of U.S. adults who are not active at all is between 23% and 27%

**Correct Interpretation #2:**We are 95% confident that the true proportion of U.S. adults who are not active at all is covered by the interval (23%, 27%)

**A More Thorough Interpretation:**Based upon our sample, the true proportion of U.S. adults who are not active at all is estimated to be 25%. With 95% confidence, this value could be as small as 23% to as large as 27%.

**A Common Interpretation in Journal Articles:**Based upon our sample, the true proportion of U.S. adults who are not active at all is estimated to be 25% (95% CI 23%-27%).

Now let’s look at an INCORRECT interpretation which we have seen before

**INCORRECT Interpretation:***There is a 95% chance that the true proportion of U.S. adults who are not active at all is between 23% and 27%*. We know this is incorrect because at this point, the true proportion and the numbers in our interval are fixed. The probability is either 1 or 0 depending on whether the interval is one of the 95% that cover the true proportion, or one of the 5% that do not.

For confidence intervals regarding a population mean, we have an additional caution to discuss about interpretations.

**Example: **Suppose a 95% confidence interval for the average minutes per day of exercise for U.S. adults is (12, 18).

**Correct Interpretation:**We are 95% confident that the true mean minutes per day of exercise for U.S. adults is between 12 and 18 minutes.

**INCORRECT Interpretation:**We are 95% confident that an individual U.S. adult exercises between 12 and 18 minutes per day. We must remember that our intervals are about the parameter, in this case the population mean. They do not apply to an individual as we expect individuals to have much more variation.

**INCORRECT Interpretation:**We are 95% confident that U.S. adults exercise between 12 and 18 minutes per day.This interpretation is implying this is true for all U.S. adults. This is an incorrect interpretation for the same reason as the previous incorrect interpretation!

As we continue to study inferential statistics, we will see that confidence intervals are used in many situations. The goal is always to provide confidence in our interval estimate of a quantity of interest. Population means and proportions are common parameters, however, any quantity that can be estimated from data has a population counterpart which we may wish to estimate.

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As we mentioned in the introduction to Unit 4A, when the variable that we’re interested in studying in the population is **categorical**, the parameter we are trying to infer about is the **population proportion (p)** associated with that variable. We also learned that the point estimator for the population proportion p is the sample proportion p-hat.

To refresh your memory, here is a picture that summarizes an example we looked at.

We are now moving on to interval estimation of p. In other words, we would like to develop a set of intervals that, with different levels of confidence, will capture the value of p. We’ve actually done all the groundwork and discussed all the big ideas of interval estimation when we talked about interval estimation for μ (mu), so we’ll be able to go through it much faster. Let’s begin.

Recall that the general form of any confidence interval for an unknown parameter is:

* estimate* ±

Since the unknown parameter here is the population proportion p, the point estimator (as I reminded you above) is the sample proportion p-hat. The confidence interval for p, therefore, has the form:

(Recall that m is the notation for the margin of error.) The margin of error (m) gives us the maximum estimation error with a certain confidence. In this case it tells us that p-hat is different from p (the parameter it estimates) by no more than m units.

From our previous discussion on confidence intervals, we also know that the margin of error is the product of two components:

To figure out what these two components are, we need to go back to a result we obtained in the Sampling Distributions section of the Probability unit about the sampling distribution of p-hat. We found that under certain conditions (which we’ll come back to later), p-hat has a normal distribution with mean p, and a

This result makes things very simple for us, because it reveals what the two components are that the margin of error is made of:

- Since, like the sampling distribution of x-bar, the sampling distribution of p-hat is normal, the confidence multipliers that we’ll use in the confidence interval for p will be the same z* multipliers we use for the confidence interval for μ (mu) when σ (sigma) is known (using
**exactly**the same reasoning and the same probability results). The multipliers we’ll use, then, are:**1.645, 1.96, and 2.576 at the 90%, 95% and 99% confidence levels, respectively.**

- The standard deviation of our estimator p-hat is

Putting it all together, we find that the confidence interval for p should be:

We just have to solve one practical problem and we’re done. We’re trying to estimate the **unknown** population proportion *p*, so having it appear in the confidence interval doesn’t make any sense. To overcome this problem, we’ll do the obvious thing …

We’ll replace p with its sample counterpart, p-hat, and work with the **estimated standard error of p-hat**

Now we’re done. The **confidence interval for the population proportion p** is:

The drug Viagra became available in the U.S. in May, 1998, in the wake of an advertising campaign that was unprecedented in scope and intensity. A Gallup poll found that by the end of the first week in May, 643 out of a random sample of 1,005 adults were aware that Viagra was an impotency medication (based on “Viagra A Popular Hit,” a Gallup poll analysis by Lydia Saad, May 1998).

Let’s estimate the proportion p of all adults in the U.S. who by the end of the first week of May 1998 were already aware of Viagra and its purpose by setting up a 95% confidence interval for p.

We first need to calculate the sample proportion p-hat. Out of 1,005 sampled adults, 643 knew what Viagra is used for, so p-hat = 643/1005 = 0.64

Therefore, a 95% confidence interval for p is

We can be 95% confident that the proportion of all U.S. adults who were already familiar with Viagra by that time was between 0.61 and 0.67 (or 61% and 67%).

The fact that the margin of error equals 0.03 says we can be 95% confident that unknown population proportion p is within 0.03 (3%) of the observed sample proportion 0.64 (64%). In other words, we are 95% confident that 64% is “off” by no more than 3%.

**Comment:**

- We would like to share with you the methodology portion of the official poll release for the Viagra example. We hope you see that you now have the tools to understand how poll results are analyzed:

“The results are based on telephone interviews with a randomly selected national sample of 1,005 adults, 18 years and older, conducted May 8-10, 1998. For results based on samples of this size, one can say with 95 percent confidence that the error attributable to sampling and other random effects could be plus or minus 3 percentage points. In addition to sampling error, question wording and practical difficulties in conducting surveys can introduce error or bias into the findings of public opinion polls.”

The purpose of the next activity is to provide guided practice in calculating and interpreting the confidence interval for the population proportion p, and drawing conclusions from it.

Two important results that we discussed at length when we talked about the confidence interval for μ (mu) also apply here:

1. There is a trade-off between level of confidence and the width (or precision) of the confidence interval. The more precision you would like the confidence interval for p to have, the more you have to pay by having a lower level of confidence.

2. Since n appears in the denominator of the margin of error of the confidence interval for p, for a fixed level of confidence, the larger the sample, the narrower, or more precise it is. This brings us naturally to our next point.

Just as we did for means, when we have some level of flexibility in determining the sample size, we can set a desired margin of error for estimating the population proportion and find the sample size that will achieve that.

For example, a final poll on the day before an election would want the margin of error to be quite small (with a high level of confidence) in order to be able to predict the election results with the most precision. This is particularly relevant when it is a close race between the candidates. The polling company needs to figure out how many eligible voters it needs to include in their sample in order to achieve that.

Let’s see how we do that.

(**Comment:** For our discussion here we will focus on a 95% confidence level (z* = 1.96), since this is the most commonly used level of confidence.)

The confidence interval for p is

The margin of error, then, is

Now we isolate n (i.e., express it as a function of m).

There is a practical problem with this expression that we need to overcome.

Practically, you first determine the sample size, then you choose a random sample of that size, and then use the collected data to find p-hat.

So the fact that the expression above for determining the sample size depends on p-hat is problematic.

The way to overcome this problem is to take the conservative approach by setting p-hat = 1/2 = 0.5.

Why do we call this approach conservative?

It is conservative because the expression that appears in the numerator,

is maximized when p-hat = 1/2 = 0.5.

That way, the n we get will work in giving us the desired margin of error regardless of what the value of p-hat is. This is a “worst case scenario” approach. So when we do that we get:

In general, for any confidence level we have

- If we know a reasonable estimate of the proportion we can use:

- If we choose the conservative estimate assuming we know nothing about the true proportion we use:

It seems like media polls usually use a sample size of 1,000 to 1,200. This could be puzzling.

How could the results obtained from, say, 1,100 U.S. adults give us information about the entire population of U.S. adults? 1,100 is such a tiny fraction of the actual population. Here is the answer:

What sample size n is needed if a margin of error m = 0.03 is desired?

(remember, always round up). In fact, 0.03 is a very commonly used margin of error, especially for media polls. For this reason, most media polls work with a sample of around 1,100 people.

As we mentioned before, one of the most important things to learn with any inference method is the conditions under which it is safe to use it.

As we did for the mean, the assumption we made in order to develop the methods in this unit was that the sampling distribution of the sample proportion, p-hat is roughly normal. Recall from the Probability unit that the conditions under which this happens are that

Since p is unknown, we will replace it with its estimate, the sample proportion, and set

to be the conditions under which it is safe to use the methods we developed in this section.

Here is one final practice for these confidence intervals!!

In general, a confidence interval for the unknown population proportion (p) is

where z* is 1.645 for 90% confidence, 1.96 for 95% confidence, and 2.576 for 99% confidence.

To obtain a desired margin of error (m) in a confidence interval for an unknown population proportion, a conservative sample size is

If a reasonable estimate of the true proportion is known, the sample size can be calculated using

The methods developed in this unit are safe to use as long as

]]>- When the population is normal and/or the sample is large, a confidence interval for unknown population mean μ (mu) when σ (sigma) is known is:

where z* is 1.645 for 90% confidence, 1.96 for 95% confidence, and 2.576 for 99% confidence.

- There is a trade-off between the level of confidence and the precision of the interval estimation. For a given sample size, the price we have to pay for more precision is sacrificing level of confidence.

- The general form of confidence intervals is an estimate +/- the margin of error (m). In this case, the estimate = x-bar and

The confidence interval is therefore centered at the estimate and its width is exactly 2m.

- For a given level of confidence, the width of the interval depends on the sample size. We can therefore do a sample size calculation to figure out what sample size is needed in order to get a confidence interval with a desired margin of error m, and a certain level of confidence (assuming we have some flexibility with the sample size). To do the sample size calculation we use:

(and round **up** to the next integer). We estimate σ (sigma) when necessary.

- When σ (sigma) is unknown, we use the sample standard deviation, s, instead, but as a result we also need to use a different set of confidence multipliers (t*) associated with the t distribution. We will use software to calculate intervals in this case, however, the formula for confidence interval in this case is

- These new multipliers have the added complexity that they depend not only on the level of confidence, but also on the sample size. Software is therefore very useful for calculating confidence intervals in this case.

- For large values of n, the t* multipliers are not that different from the z* multipliers, and therefore using the interval formula:

for μ (mu) when σ (sigma) is unknown provides a pretty good approximation.

]]>As we just learned, for a given level of confidence, the sample size determines the size of the margin of error and thus the width, or precision, of our interval estimation. This process can be reversed.

In situations where a researcher has some flexibility as to the sample size, the researcher can calculate in advance what the sample size is that he/she needs in order to be able to report a confidence interval with a certain level of confidence and a certain margin of error. Let’s look at an example.

Recall the example about the SAT-M scores of community college students.

An educational researcher is interested in estimating μ (mu), the mean score on the math part of the SAT (SAT-M) of all community college students in his state. To this end, the researcher has chosen a random sample of 650 community college students from his state, and found that their average SAT-M score is 475. Based on a large body of research that was done on the SAT, it is known that the scores roughly follow a normal distribution, with the standard deviation σ (sigma) =100.

The 95% confidence interval for μ (mu) is

which is roughly 475 ± 8, or (467, 483). For a sample size of n = 650, our margin of error is 8.

Now, let’s think about this problem in a slightly different way:

An educational researcher is interested in estimating μ (mu), the mean score on the math part of the SAT (SAT-M) of all community college students in his state with a margin of error of (only) 5, at the 95% confidence level. What is the sample size needed to achieve this? σ (sigma), of course, is still assumed to be 100.

To solve this, we set:

So, for a sample size of 1,600 community college students, the researcher will be able to estimate μ (mu) with a margin of error of 5, at the 95% level. In this example, we can also imagine that the researcher has some flexibility in choosing the sample size, since there is a minimal cost (if any) involved in recording students’ SAT-M scores, and there are many more than 1,600 community college students in each state.

Rather than take the same steps to isolate n every time we solve such a problem, we may obtain a general expression for the required n for a desired margin of error m and a certain level of confidence.

Since

is the formula to determine m for a given n, we can use simple algebra to express n in terms of m (multiply both sides by the square root of n, divide both sides by m, and square both sides) to get

**Comment:**

- Clearly, the
**sample size n must be an integer**. - In the previous example we got n = 1,600, but in other situations, the calculation may give us a non-integer result.
- In these cases, we should always
**round up to the next highest integer.** - Using this “conservative approach,” we’ll achieve an interval at least as narrow as the one desired.

IQ scores are known to vary normally with a standard deviation of 15. How many students should be sampled if we want to estimate the population mean IQ at 99% confidence with a margin of error equal to 2?

Round up to be safe, and take a sample of 374 students.

The purpose of the next activity is to give you guided practice in sample size calculations for obtaining confidence intervals with a desired margin of error, at a certain confidence level. Consider the example from the previous Learn By Doing activity:

**Comment:**

- In the preceding activity, you saw that in order to calculate the sample size when planning a study, you needed to know the population standard deviation, sigma (σ). In practice, sigma is usually not known, because it is a parameter. (The rare exceptions are certain variables like IQ score or standardized tests that might be constructed to have a particular known sigma.)

Therefore, when researchers wish to compute the required sample size in preparation for a study, they use an **estimate** of sigma. Usually, sigma is estimated based on the standard deviation obtained in prior studies.

However, in some cases, there might not be any prior studies on the topic. In such instances, a researcher still needs to get a rough estimate of the standard deviation of the (yet-to-be-measured) variable, in order to determine the required sample size for the study. One way to get such a rough estimate is with the “range rule of thumb.” We will not cover this topic in depth but mention here that a very rough estimate of the standard deviation of a population is the range/4.

There are a few more things we need to discuss:

- Is it always OK to use the confidence interval we developed for μ (mu) when σ (sigma) is known?

- What if σ (sigma) is unknown?

- How can we use statistical software to calculate confidence intervals for us?

One of the most important things to learn with any inference method is the conditions under which it is safe to use it. It is very tempting to apply a certain method, but if the conditions under which this method was developed are not met, then using this method will lead to unreliable results, which can then lead to wrong and/or misleading conclusions. As you’ll see throughout this section, we will always discuss the conditions under which each method can be safely used.

In particular, the confidence interval for μ (mu), when σ (sigma) is known:

was developed assuming that the sampling distribution of x-bar is normal; in other words, that the Central Limit Theorem applies. In particular, this allowed us to determine the values of z*, the confidence multiplier, for different levels of confidence.

First, **the sample must be random.** Assuming that the sample is random, recall from the Probability unit that the Central Limit Theorem works when the **sample size is large** (a common rule of thumb for “large” is n > 30), or, for **smaller sample sizes**, if it is known that the quantitative **variable** of interest is **distributed normally** in the population. The only situation when we cannot use the confidence interval, then, is when the sample size is small and the variable of interest is not known to have a normal distribution. In that case, other methods, called non-parametric methods, which are beyond the scope of this course, need to be used. This can be summarized in the following table:

In the following activity, you have to opportunity to use software to summarize the raw data provided.

As we discussed earlier, when variables have been well-researched in different populations it is reasonable to assume that the population standard deviation (σ, sigma) is known. However, this is rarely the case. What if σ (sigma) is unknown?

Well, there is some good news and some bad news.

The good news is that we can easily replace the population standard deviation, σ (sigma), with the **sample** standard deviation, s.

The bad news is that once σ (sigma) has been replaced by s, we lose the Central Limit Theorem, together with the normality of x-bar, and therefore the confidence multipliers z* for the different levels of confidence (1.645, 1.96, 2.576) are (generally) not correct any more. The new multipliers come from a different distribution called the “t distribution” and are therefore denoted by t* (instead of z*). We will discuss the t distribution in more detail when we talk about hypothesis testing.

The confidence interval for the population mean (μ, mu) when (σ, sigma) is unknown is therefore:

(Note that this interval is very similar to the one when σ (sigma) is known, with the obvious changes: s replaces σ (sigma), and t* replaces z* as discussed above.)

There is an important difference between the confidence multipliers we have used so far (z*) and those needed for the case when σ (sigma) is unknown (t*). Unlike the confidence multipliers we have used so far (z*), which depend only on the level of confidence, the new multipliers (t*) have the **added complexity** that they **depend on both the level of confidence and on the sample size** (for example: the t* used in a 95% confidence when n = 10 is different from the t* used when n = 40). Due to this added complexity in determining the appropriate t*, we will rely heavily on software in this case.

**Comments:**

- Since it is quite rare that σ (sigma) is known, this interval (sometimes called a “one-sample t confidence interval”) is more commonly used as the confidence interval for estimating μ (mu). (Nevertheless, we could not have presented it without our extended discussion up to this point, which also provided you with a solid understanding of confidence intervals.)

- The quantity s/sqrt(n) is called the
**estimated standard error**of x-bar. The Central Limit Theorem tells us that σ/sqrt(n) = sigma/sqrt(n) is the**standard deviation**of x-bar (and this is the quantity used in confidence interval when σ (sigma) is known). In general, the**standard error**is the**standard deviation of the sampling distribution of a statistic**. When we substitute s for σ (sigma) we are estimating the true standard error. You may see the term “standard error” used for both the true standard error and the estimated standard error depending on the author and audience. What is important to understand about the standard error is that it measures the variation of a statistic calculated from a sample of a specified sample size (not the variation of the original population).

- As before, to safely use this confidence interval (one-sample t confidence interval), the sample
**must be random**, and the only case when this interval cannot be used is when the sample size is small and the variable is not known to vary normally.

**Final Comment:**

- It turns out that for large values of n, the t* multipliers are not that different from the z* multipliers, and therefore using the interval formula:

for μ (mu) when σ (sigma) is unknown provides a pretty good approximation.

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