View Lecture Slides with Transcript – Population Means – Part 3

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View Lecture Slides with Transcript – Population Means – Part 1

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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.

- 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.

]]>The purpose of this activity is to use statistical software for calculating the sample mean. Software is particularly useful when all you have are the raw data (no summary has been calculated), which is what you encounter in practice. In all the examples and activities we looked at so far, the sample mean is given (rather than the whole data set), in which case it will often take you less time to calculate the confidence interval by hand than to launch a software program and ask it to do the calculation for you. To answer these questions, you will need to recall how to find the sample mean using software.

**Background:** Some studies suggest that women having their first baby at age 35 or older are at increased risk of having a baby with a low birth weight. A medical researcher wanted to estimate μ (mu), the mean weight of newborns who are the first child for women over the age of 35. To this end, the researcher chose a random sample of 125 women age 35 or older who were pregnant with their first child and followed them through the pregnancy. The datafile linked below contains the birth weight (in grams) of the 125 newborns (women pregnant with more than one child were excluded from the study). From past research, it is assumed that the weight of newborns has a standard deviation of σ = sigma = 500 grams. We will estimate μ (mu) with a 99% confidence interval.

The dataset is available in Excel format (birthweight.xls) and in CSV format (birthweight.csv).

If you prefer to skip this process and just answer the questions, you can reveal the sample mean for this data.

This document is linked from Population Means (Part 3).

]]>**Situation A:** In order to estimate μ (mu), the mean annual salary of high-school teachers in a certain state, a random sample of 150 teachers was chosen and their average salary was found to be $38,450. From past experience, it is known that teachers’ salaries have a standard deviation of $5,000.

**Situation B: **A medical researcher wanted to estimate μ (mu), the mean recovery time from open-heart surgery for males between the ages of 50 and 60. The researcher followed the next 15 male patients in this age group who underwent open-heart surgery in his medical institute through their recovery period. (Comment: Even though the sample was not strictly random, there is no reason to believe that the sample of “the next 15 patients” introduces any bias, so it is as good as a random sample). The mean recovery time of the 15 patients was 26 days. From the large body of research that was done in this area, it is assumed that recovery times from open-heart surgery have a standard deviation of 3 days.

**Situation C: **In order to estimate μ (mu), the mean score on the quantitative reasoning part of the GRE (Graduate Record Examination) of all MBA students, a random sample of 1,200 MBA students was chosen, and their scores were recorded. The sample mean was found to be 590. It is known that the quantitative reasoning scores on the GRE vary normally with a standard deviation of 150.

**Situation D: **A psychologist wanted to estimate μ (mu), the mean time it takes 6-year-old children diagnosed with Down’s Syndrome to complete a certain cognitive task. A random sample of 12 children was chosen and their times were recorded. The average time it took the 12 children to complete the task was 7.5 minutes. From past experience with similar tasks, the time is known to vary normally with a standard deviation of 1.3 minutes.

Here is another set of situations:

**Situation A:** A marketing executive wants to estimate the average time, in days, that a watch battery will last. She tests 50 randomly selected batteries and finds that the distribution is skewed to the left, since a couple of the batteries were defective. It is known from past experience that the standard deviation is 25 days.

**Situation B:** A college professor desires an estimate of the mean number of hours per week that full-time college students are employed. He randomly selected 250 college students and found that they worked a mean time of 18.6 hours per week. He uses previously known data for his standard deviation.

**Situation C:** A medical researcher at a sports medicine clinic uses 35 volunteers from the clinic to study the average number of hours the typical American exercises per week. It is known that hours of exercise are normally distributed and past data give him a standard deviation of 1.2 hours.

**Situation D:** A high-end auto manufacturer tests 5 randomly selected cars to find out the damage caused by a 5 mph crash. It is known that this distribution is normal. Assume that the standard deviation is known.

This document is linked from Population Means (Part 3).

]]>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.

]]>In the previous activity, we calculated a 95% confidence interval for μ (mu), the mean pregnancy length of women who smoke during their pregnancy based on the given information, and found it to be 260 +/- 3, or (257, 263).

This document is linked from Population Means (Part 3).

]]>95% is the most commonly used level of confidence. However, we may wish to increase our level of confidence and produce an interval that’s almost certain to contain μ (mu). Specifically, we may want to report an interval for which we are 99% confident that it contains the unknown population mean, rather than only 95%.

Using the same reasoning as in the last comment, in order to create a 99% confidence interval for μ (mu), we should ask: There is a probability of 0.99 that any normal random variable takes values within how many standard deviations of its mean? The precise answer is 2.576, and therefore, a 99% confidence interval for μ (mu) is:

Another commonly used level of confidence is a 90% level of confidence. Since there is a probability of 0.90 that any normal random variable takes values within 1.645 standard deviations of its mean, the 90% confidence interval for μ (mu) is:

Let’s go back to our first example, the IQ example:

The IQ level of students at a particular university has an unknown mean (μ, mu) and known standard deviation σ (sigma) =15. A simple random sample of 100 students is found to have a sample mean IQ of 115 (x-bar). Estimate μ (mu) with a 90%, 95%, and 99% confidence interval.

A 90% confidence interval for μ (mu) is:

A 95% confidence interval for μ (mu) is:

A 99% confidence interval for μ (mu) is:

The purpose of this next activity is to give you guided practice at calculating and interpreting confidence intervals, and drawing conclusions from them.

Note from the previous example and the previous “Did I Get This?” activity, that the more confidence I require, the wider the confidence interval for μ (mu). The 99% confidence interval is wider than the 95% confidence interval, which is wider than the 90% confidence interval.

This is not very surprising, given that in the 99% interval we multiply the standard deviation of the statistic by 2.576, in the 95% by 2, and in the 90% only by 1.645. Beyond this numerical explanation, there is a very clear intuitive explanation and an important implication of this result.

Let’s start with the intuitive explanation. The more certain I want to be that the interval contains the value of μ (mu), the more plausible values the interval needs to include in order to account for that extra certainty. I am 95% certain that the value of μ (mu) is one of the values in the interval (112.1, 117.9). In order to be 99% certain that one of the values in the interval is the value of μ (mu), I need to include more values, and thus provide a wider confidence interval.

In our example, the **wider** 99% confidence interval (111, 119) gives us a **less precise** estimation about the value of μ (mu) than the narrower 90% confidence interval (112.5, 117.5), because the smaller interval ‘narrows-in’ on the plausible values of μ (mu).

The important practical implication here is that researchers must decide whether they prefer to state their results with a higher level of confidence or produce a more precise interval. In other words,

The price we have to pay for a higher level of confidence is that the unknown population mean will be estimated with less precision (i.e., with a wider confidence interval). If we would like to estimate μ (mu) with more precision (i.e. a narrower confidence interval), we will need to sacrifice and report an interval with a lower level of confidence.

So far we’ve developed the confidence interval for the population mean “from scratch” based on results from probability, and discussed the trade-off between the level of confidence and the precision of the interval. The price you pay for a higher level of confidence is a lower level of precision of the interval (i.e., a wider interval).

Is there a way to bypass this trade-off? In other words, is there a way to increase the precision of the interval (i.e., make it narrower) **without **compromising on the level of confidence? We will answer this question shortly, but first we’ll need to get a deeper understanding of the different components of the confidence interval and its structure.

We explored the confidence interval for μ (mu) for different levels of confidence, and found that in general, it has the following form:

where z* is a general notation for the multiplier that depends on the level of confidence. As we discussed before:

- For a 90% level of confidence, z* = 1.645

- For a 95% level of confidence, z* = 1.96

- For a 99% level of confidence, z* = 2.576

To start our discussion about the structure of the confidence interval, let’s denote

The confidence interval, then, has the form:

To summarize, we have

X-bar is the sample mean, the point estimator for the unknown population mean (μ, mu).

**m** is called the **margin of error**, since it represents the maximum estimation error for a given level of confidence.

For example, for a 95% confidence interval, we are 95% confident that our estimate will not depart from the true population mean by more than m, the margin of error and m is further made up of the product of two components:

Here is a summary of the different components of the confidence interval and its structure:

This structure: **estimate ± margin of error**, where the margin of error is further composed of the product of a confidence multiplier and the standard deviation of the statistic (or, as we’ll see, the standard error) is the general structure of all confidence intervals that we will encounter in this course.

Obviously, even though each confidence interval has the same components, the formula for these components is different from confidence interval to confidence interval, depending on what unknown parameter the confidence interval aims to estimate.

Since the structure of the confidence interval is such that it has a margin of error on either side of the estimate, it is centered at the estimate (in our current case, x-bar), and its width (or length) is exactly twice the margin of error:

The margin of error, m, is therefore “in charge” of the width (or precision) of the confidence interval, and the estimate is in charge of its location (and has no effect on the width).

Let us now go back to the confidence interval for the mean, and more specifically, to the question that we posed at the beginning of the previous page:

Is there a way to increase the precision of the confidence interval (i.e., make it narrower) **without** compromising on the level of confidence?

Since the width of the confidence interval is a function of its margin of error, let’s look closely at the margin of error of the confidence interval for the mean and see how it can be reduced:

Since z* controls the level of confidence, we can rephrase our question above in the following way:

Is there a way to reduce this margin of error other than by reducing z*?

If you look closely at the margin of error, you’ll see that the answer is **yes.** We can do that by increasing the sample size n (since it appears in the denominator).

**Question :** Isn’t it true that another way to reduce the margin of error (for a fixed z*) is to reduce σ (sigma)?

**Answer: **While it is true that strictly mathematically speaking the smaller the value of σ (sigma), the smaller the margin of error, practically speaking we have absolutely no control over the value of σ (sigma) (i.e., we cannot make it larger or smaller). σ (sigma) is the population standard deviation; it is a fixed value (which here we assume is known) that has an effect on the width of the confidence interval (since it appears in the margin of error), but is definitely not a value we can change.

Let’s look at an example first and then explain why increasing the sample size is a way to increase the precision of the confidence interval **without **compromising on the level of confidence.

Recall the IQ example:

The IQ level of students at a particular university has an unknown mean (μ, mu) and a known standard deviation of σ (sigma) =15. A simple random sample of 100 students is found to have the sample mean IQ of 115 (x-bar).

For simplicity, in this question, we will round z* = 1.96 to 2. You should use z* = 1.96 in all problems unless you are specifically instructed to do otherwise.

A 95% confidence interval for μ (mu) in this case is:

Note that the margin of error is m = 3, and therefore the width of the confidence interval is 6.

Now, what if we change the problem slightly by increasing the sample size, and assume that it was 400 instead of 100?

In this case, a 95% confidence interval for μ (mu) is:

The margin of error here is only m = 1.5, and thus the width is only 3.

Note that for the same level of confidence (95%) we now have a narrower, and thus more precise, confidence interval.

Let’s try to understand why is it that a larger sample size will reduce the margin of error for a fixed level of confidence. There are three ways to explain this: mathematically, using probability theory, and intuitively.

We’ve already alluded to the **mathematical** explanation; the margin of error is

and since n, the sample size, appears in the denominator, increasing n will reduce the margin of error.

As we saw in our discussion about point estimates, **probability theory** tells us that:

This explains why with a larger sample size the margin of error (which represents how far apart we believe x-bar might be from μ (mu) for a given level of confidence) is smaller.

On an intuitive level, if our estimate x-bar is based on a larger sample (i.e., a larger fraction of the population), we have more faith in it, or it is more reliable, and therefore we need to account for less error around it.

**Comment:**

- While it is true that for a given level of confidence, increasing the sample size increases the precision of our interval estimation, in practice, increasing the sample size is not always possible.
- Consider a study in which there is a non-negligible cost involved for collecting data from each participant (an expensive medical procedure, for example). If the study has some budgetary constraints, which is usually the case, increasing the sample size from 100 to 400 is just not possible in terms of cost-effectiveness.
- Another instance in which increasing the sample size is impossible is when a larger sample is simply not available, even if we had the money to afford it. For example, consider a study on the effectiveness of a drug on curing a very rare disease among children. Since the disease is rare, there are a limited number of children who could be participants.

- This is the reality of statistics. Sometimes theory collides with reality, and you simply do the best you can.