Hypothesis Testing:
Continuous Variables (1 Sample)

1. Sampling Distribution of the Mean
   1. Empirical
   2. Theoretical
   3. Sampling Distributions & Normality
2. Case I - Parameters Known
   1. Rationale
   2. Formal Example - [Minitab]
      1. Research Question
      2. Hypotheses
      3. Assumptions
      4. Decision Rules
      5. Computation
      6. Decision
3. Errors & the Power of a Test
4. Case II - Sigma Unknown
   1. Rationale
   2. Formal Example - [Minitab]
      1. Research Question
      2. Hypotheses
      3. Assumptions
      4. Decision Rules
      5. Computation
      6. Decision
5. Interval Estimation

Homework

I. Sampling Distribution of the Mean

As might be expected, inference with continuous variables is more complicated than with dichotomous variables. Fortunately however, the general principles are the same. Again, we will use a sampling distribution to index the probability that the observed outcome is due to chance.

1. Empirical (capable of being verified or disproved by observation or experiment)
   The sampling distribution of the mean is a probability distribution of the possible values of the mean that would occur if we were to draw all possible samples of a fixed size from a given population.

   To get a better feel for this notion, let’s consider an empirical example (or one that could be actually performed). Let us choose 10 samples of size 4 from a population of size 20.
   
   

   Population Distribution	10 observed Sample Distributions	Empirical Sampling Distribution of the Mean
   6,  2, 9, 5, 0, 1, 3, 2, 1, 1, 5, 2, 7, 7, 7, 8, 1, 1, 3, 7	1, 5, 9, 0	3.75 (& sX=4.11)
   	0, 3, 1, 5	2.25
   	5, 8, 3, 0	4.00
   	1, 5, 0, 7	3.25
   	7, 6, 1, 3	4.25
   	3, 2, 1, 7	3.25
   	2, 0, 3, 5	2.50
   	1, 2, 1, 1	1.25
   	2, 7, 1, 7	4.25
   	9, 7, 6, 2	6.00
   	Ns=4

   Notes:

   1. 
   2. 
   3. There are three types of distributions illustrated above: population, sample, and sampling.
   4. Empirical sampling distributions are only used to help students understand the concept. They are not true sampling distributions, since all possible samples are not chosen.
      
      
2. Theoretical
   Are called theoretical because all possible samples (an infinite number) should be drawn. Since this is impossible, the characteristics (i.e., the mean & standard deviation) of the distribution are determined mathematically.

   It turns out that (note that and are synonyms).

   The standard deviation of the distribution of sample means is called the standard error of the mean (or more simply, the standard error). It measures variability in the distribution of sample means or, in other words, sampling error (the amount of error we can expect due to using a sample mean to estimate a population mean). One would expect the size of the standard error to be related to the sample size, and it is.

   When population values are known:

   
   Thus, as the sample size gets bigger, sampling error gets smaller.

   When population values are estimated from sample values:

   
   This formula requires s_x to be an unbiased estimator of s_x

   Computational formula for the standard error estimated from sample values:

   

   Example (using the population distribution from the empirical sampling distribution above):

   

   If we didn’t know the population values, we could use the S_X from the first sample.

   

   As you can see, only estimates and it does so poorly in this case (because of the small sample size).

3. Sampling Distributions & Normality
   The techniques that we are discussing require that the sampling distribution (in this case the distribution of sample means) be normal in shape. This will be the case if either of the following two conditions are met.
   1. The population distribution of raw scores is normal.
      It is difficult to actually know this, but fortunately, many variables are.

   2. The sampling distribution will approach normality as the sample size is increased.
      This occurs even though the population distribution may not be normal in shape. Note, though, that the more skewed the population distribution, the larger the N (sample size) needed for the sampling distribution of the mean to be normal.

II. Case I - Parameters Known

1. Rationale
   Now that we have an understanding of sampling distributions of a continuous variable, we can go on to test a hypothesis. Recall that any normally distributed variable can be transformed into a standard normal distribution (i.e., z scores). We also saw that area under the curve implies probability. Thus, if the sampling distribution of the mean is normal we can establish the probability of obtaining a particular sample mean.
   
   
2. Formal Example - [Minitab]
   Let us look at an example that will set the stage for the format we will use when testing a hypothesis. We will use one from your book. Animal studies suggest that the anticholinergic drug physostigmine improves memory. This could have some clinical applications in humans (e.g., senility, Alzheimer’s disease). Studies with humans typically report that we remember an average of seven of 15 words given an 80-minute retention interval. These studies also suggest a standard deviation for the population of two.

1. Research Question
   Does physostigmine improve memory in humans?
   
   
2. Hypotheses
   

	In Symbols	In Words
HO	m=7	Physostigmine has no effect on memory.
HA	m¹7	Physostigmine has an effect on memory.

These hypotheses should be specific to what is being tested.

3. Assumptions
   
   1. Sample is randomly selected.
   2. Population of non-drugged folks is normal.
      Reason is so that the sampling distribution of the mean will be normal. Although a large sample size would also produce a normally shaped sampling distribution, we will rarely use large samples.
   3. Population of non-drugged folks has m=7 and s=2 (i.e., the null).
      
      

4. Decision Rules
   We will use the standard normal curve (Z scores) to obtain the probabilities. Our alpha level is .05 with a two-tailed test. When we look in a Z table, we see that the critical value of Z is 1.96 (Z_crit).

   

   Thus, the shaded area is the critical region. If our observed z value falls into this area, we will reject the null hypothesis. More formally:

   If Z_obs £ -1.96 or Z_obs ³ 1.96, then reject H_O.
   If Z_obs > -1.96 and Z_obs < 1.96, then do not reject H_O.

5. Computation
   The computations have two goals corresponding to the descriptive and inferential statistics. Suppose we obtain the following scores for a sample of 20 subjects:
   
   

   9	8	8	9	9
   8	10	8	10	7
   7	7	8	8	10
   9	8	8	7	9

   
   The first step is to describe the data. The most important descriptive statistic in this case is the mean or average number of words remembered by the 20 subjects receiving the drug. The calculation reveals a mean of 8.35, which is greater the the mean of the population of 7. The second step of the computation is to perform an inferential test to determine whether this difference between means is worth paying attention to (in other words, is the improvement in memory due to sampling error or to the drug?).

   Remember:

   

   More generally:

   

   Thus:

   

   And:

   

6. Decision
   Since 3.02 (Z_obs) > 1.96 (Z_crit) we reject H_O and assert the alternative. Now we must go beyond this simple decision of rejecting the null or not to what it all means. In other words, we need to make a conclusion based on our decision and the particular results observed. In this case, we would conclude that the physostigmine improves memory. Notice that we have actually gone beyond the alternative hypothesis by specifying that the effect has a direction (memory was improved). We do this because the mean words remembered for the drugged group was higher than for the population.

III. Errors & the Power of a Test

As can be seen, hypothesis testing is just educated guessing. Moreover, guesses (educated or not) are sometimes wrong. Consider the possible decisions we can make:

Possibilities:		Actual Situation
		HO is True	HO is False
Decision	Reject HO	Type I Error	Correct Decision II
	Do not reject HO	Correct Decision I	Type II Error

Let us now consider each decision in more detail.

A Type I Error is the false rejection of a true null. It has a probability of alpha (a). In other words, this error occurs as a result of the fact that we have to somehow separate probable from improbable.

Correct Decision I occurs when we fail to reject a true null. It has a probability of 1-a. From a scientist's perspective this is a "boring" result.

A Type II Error is the false retention of a false null. It has a probability equal to beta (b).

Correct Decision II occurs when we reject a false null. The whole purpose of the experiment is to provide the occasion for this type of decision. In other words, we performed the statistical test because we expect the sample to differ. This decision has a probability of 1-b. This probability is also known as the power of the statistical test. In other words, the ability of a test to find a difference when there really is one, is power.

Factors Influencing Power:

1. Alpha (a). Alpha and beta are inversely related. In other words, as one increases, the other decreases (i.e., a ´ b = K). Thus, all other things being equal, using an alpha of .05 will result in a more powerful test than using an alpha of .01.

2. Sample Size (N). The bigger the sample (i.e., the more work we do), the more powerful the test.

3. Type of Test. Metric tests (as compared to nonparametric tests that we discuss later in the semester) are generally more powerful due to assumptions that are more restrictive.

4. Variability. Generally speaking, variability in the sample and/or population results in a less powerful test.
   
   
5. Test Directionality. One-tailed tests have the potential to be more powerful than two-tailed tests.
   
   
6. Robustness of the Effect. Six beers are more likely to influence reaction time than one beer.
   
   

IV. Case II - Sigma Unknown

In the Case I example, both the mean (m) and standard deviation (s) of the population were given. However, these parameters are rarely known. In this section, we will consider how the test is performed when s is unknown.

1. Rationale

As we noted earlier, can be used to estimate . One complication of doing this is that the shape of the theoretical distribution of sample means will depend on the sample size. Thus, this sampling distribution is actually a family of distributions and is called Student’s t. To better understand the t distributions, we need to consider a new way of thinking of sample size.

The Degrees of Freedom (df) for a statistic refer to the number of calculations in its computation that are free to vary. For example, the df for the variance of a sample (S_x²) is N-1.

In other words, since the sum of the deviations equals zero, N-1 of the deviations are free to vary. That is, given N-1 of the deviations, we can easily determine the final deviation because it is not free to vary. In the example below where N=5, the unknown value must be 2.

c
-2
-1
0
1
?
åc=0

In the Case II situation, the df for t equals the df for S_x which is N-1. And Student’s t is a family of distributions differing in their kurtosis (or peakedness).

Note that when df are infinite (i.e., the sample size is very large), the t distribution will equal the z distribution.

As for the formula, remember the z test:

The formula for the t is similar.

Like the z test, the critical values of t are obtained from a table. To determine the critical value of t from the table, you will need to know a, the df, and whether you are using a one- or two-tailed test.

2. Formal Example - [Minitab]

You are interested in whether the average IQ for a group of "bad kids" (the ones that put a tack on your seat before you sit down) in a school is different from the rest of the kids in the school. The average IQ for the school as a whole is 102 with the standard deviation unavailable.

1. Research Question
   Do "bad kids" have normal intelligence?
   
   
2. Hypotheses
   

   	In Symbols	In Words
   HO	m=102	Bad kids have normal IQs.
   HA	m¹102	Bad kids do not have normal IQs.

   
   
3. Assumptions
   

1. Sample is randomly selected.
2. Population of IQ is normal.
   Reason is so that the sampling distribution of the mean will be normal. Although a large sample size would also produce a normally shaped sampling distribution, we will rarely use large samples.
3. Population of IQ has m=102.
   
   
4. Decision Rules
   Using alpha of .05 with a two-tailed test and N=20 (df=N-1=19), we determine from the t table that the critical value is 2.093.
   
   
   
   Thus:

If t_obs £ -2.093 or t_obs ³ 2.093, then reject H_O.
If t_obs > -2.093 and t_obs < 2.093, then do not reject H_O.

5. Computation
   The IQs for the 20 bad kids are as follows:
   
   

   106	111	120	88	109
   120	123	127	91	130
   118	88	97	110	92
   124	116	118	114	108

   
   Describing the data, we determine that the average IQ is 110.5 with S_X=13.12. We then go on to perform the appropriate inferential test. Thus:
   

   

   

6. Decision
   Since 2.90 (t_obs) > 2.093 (t_crit) we reject H_O and assert the alternative. In other words, we conclude that the "bad kids" are smarter than average. Notice that we have actually gone beyond the alternative hypothesis by specifying that the effect has a direction (bad kids are smarter).

V. Interval Estimation

Suppose you are a researcher interested in self-destructiveness. You develop a scale to measure this trait. Example questions might include:

1. I like to listen to loud music.
2. I use (or have used) drugs.
3. I like to drive fast.

Next you obtain a random sample of 25 people and give them the scale. (The difficulty in obtaining a random sample might be noted.) The mean for this sample is 120 and the standard deviation is 10.

One of the things that we may want to know is what is the range of scores expected for the population. If we knew this, we would be easily able to identify the deviant scorer (possibly for a case study).

Let’s say we wanted to know the expected range of scores for 95% of the population. This is termed the 95% Confidence Interval (CI) and is given by:

with df=N-1,
and remembing

Thus, the current example has df=N-1, alpha of .05 (for 95% CI), two-tailed test. From the t table, we determine that the t_crit is 2.064.

Therefore, the upper limit will be

And the lower limit will be:

We can now be confident that 95% of people would be expected to score between 115.87 and 124.13.