> 7 .KbjbjUU "7|7|GlbbbbbbbvHHH8HT&Iv
i2FJ>M"`M`M`M`M>MMhhhhhhh 30, table (1) assures us that we can compute a z-value and use a Z-test.
That sounds great, you say, but what does it mean, you sigh? A z-value is a number we compute which can then be graphed as a point on the horizontal scale of the standard normal distribution curve. This point indicates how far from the population mean our sample mean is, and thus enables us to determine how "unusual" our research finding are.
FOR YOUR INFORMATION AND EDUCATION:
The Central Limit Theorem implies that for samples of sizes larger than 30, the sample means can be approximated reasonably well by a normal (z) distribution. The approximation gets better as the sample size, n, becomes larger.
When you compute a z-value you are converting your mean to a mean of 0 and your standard deviation to a standard deviation of 1. This allows you to use the standard normal distribution curve and its' corresponding table to determine the significance of your values regardless of the actual value of your mean or standard deviation.
A standard normal probability distribution is a bell-shaped curve (also called a Gaussian curve in honor of its discoverer, Karl Gauss) where the mean, or middle value is 0, and the standard deviation, the place where the curve starts to bend, is equal to 1 on the right and -1 on the left. The area under every probability distribution curves is equal to 1 or 100%). Since a Gaussian curve is symmetric about the mean, it is important to note that the mean divides this curve into 2 equal areas of 50%.
Blood cholesterol levels, heights of adult women, weights of 10 year old boys, diameters of apples, scores on standardized test, etc., are all examples of collections of values whose frequency distribution resemble the Gaussian curve.
If you were to record all the possible outcomes from the toss of 100 different coins, by graphing the number of heads that could occur on the horizontal axis (0,1,2,3...100), and the frequency with which each of the number of heads could occur on the vertical axis, you will produce a graph resembling the normal distribution. (The most "frequent" results would cluster around 50 heads and become less and less frequent as you consider values further and further from 50.]
6. Perform the Arithmetic to find the test statistic, the critical value(s), and the critical region. [actually the computer will do this for you(]
The sample mean of 7.8 is equivalent to a z value of 2.37. This z value is the test statistic, and was computed using the following formula:
z = EMBED Equation.DSMT4
x = sample mean,
= population mean,
n = size of sample
EMBED Equation.DSMT4 = population standard deviation,
Thus, z = (7.8 - 7.5) / (0.76)/6 = 2.37 (to the nearest hundredth)
Note: z numbers usually vary between -3 and +3. If they are outside of this range the null hypothesis will almost always be rejected (or an error was made).
Note: EMBED Equation.DSMT4 (sigma divided by the square root of n) is often called the standard error of the mean or the standard deviation of the sample means.
Note: Sometimes you can substitute the actual standard deviation of the sample, s, for the population standard deviation, s, if EMBED Equation.DSMT4 is unknown.
The EMBED Equation.DSMT4 = 0.05 level employs us to find a z-value that will separate the curve into 2 unequal regions; the smaller one with an area of 0.05 (5%), and the larger one with an area of 100% - 5% = 95% (0.95). (Often referred to as a 95% confidence level).
Z-values indicate the percent of area under the bell-shaped curve from the mean (middle) towards the "right tail" of the curve. Thus, for an alpha of 0.05, we need to determine what z value will cut off an area of 45% (0.4500) from the mean towards the "right tail" (we already know that 50% of the area is on the left side of the mean. We obtain 95% by adding 45% to 50%).
"Hunting" through the vast array of four digit numerals in the table, we find our critical value to be between 1.6 (see z column) +. 04 (1.64) which (reading down the .04 column) determines an area of .4495, and 1.6 +. 05 (1.65), which determines an area of .4505. Thus, if we take the mean average of 1.64 and 1.65, we can blissfully determine the critical value to be 1.645. and the critical region to be all Z values greater than 1.645. This determination requires us to reject the null hypothesis if our test statistics (Z value) is greater than 1.645.
Note: Since there is only one alternative hypothesis, H1: EMBED Equation.DSMT4 we call this a one-tailed (right tailed) test. If our alternative hypothesis was EMBED Equation.DSMT4 we would have a left-tailed test, and a two-tailed test would be used since there would be two alternatives: EMBED Equation.DSMT4 or EMBED Equation.DSMT4 .
Note: The p value is .0089. We arrived at this value quite easily. Recall, in step 6 we computed the A relation between variables score to be 2.37. If you go down the left side of the table and find a z value of 2.3, then go across to the .07 column, you find, the number .4911. This indicates an area of 49.11% from the mean z value of 0 to the z value of 2.37. Thus, 50%+49.11% or 99.11% of the area of the curve is to the left of 2.37 and the small tail to the right of 2.37 has an area of 100%- 99.11% = 0.89% or 0.0089 which is our p-value.
7. Look to reject or fail to reject (accept) the null hypothesis. Reject null hypothesis if the test statistic is in the critical region. Fail to reject the null hypothesis if the test statistic is not in the critical region.
Ms. Rodriquez's z-value is in the critical region since 2.37 > 1.645. Thus we will reject the null hypothesis.
8. Restate the previous decision in simple Lay non-technical terms.
We have reason to believe that the new batteries reduce the time before re-charging.
FOR YOUR INFORMATION AND EDUCATION:
If your sample size is less than 30 and the population standard deviation is unknown, but there is a normal distribution in the population, then you can compute a "t" statistic, where:
t = EMBED Equation.DSMT4
x is the sample mean, is the population mean under contention, s is the standard deviation of the sample, and n is the size of the sample.
Notice that the z and t statistics are computed the same way, the only difference is in their corresponding values of significance when you (or your computer) checks these values on the graph or table.
Summary
In this example we tested a claim about a numerical value mean test using ratio data. The sample data came from a population, which was known to have a normal distribution. We were thus able to use parametric methods in our hypothesis testing.
In general, when you test claims with interval or ratio parameters (such as mean, standard deviation, or proportions), and some fairly strict requirements (such as the sample data came from a normally distributed population) are met, you should be able to use parametric methods.
If the necessary requirements for parametric methods are not met, do not despair. It is very likely that there are alternative techniques, named appropriately, non-parametric methods, which could (and should) be used instead.
Since they do not require normally distributed populations, non-parametric tests are often called distribution-free tests. Some advantages to using non-parametric methods include:
1. They can often be applied to nominal data that lack exact numerical values.
2. They usually involve computations that are simpler than the corresponding parametric methods.
3. They tend to be easier to understand.
Unfortunately, there are the following disadvantages:
1. Non-parametric methods tend to "waste" information, since exact numerical data are often reduced to a qualitative form. Some treat all values as either positive (+) or negative (-). [A dieter using this test would count a loss of 60 lbs as the same as a loss of 2 lbs!].
2. They are generally less sensitive than the corresponding parametric methods. This means that you need stronger evidence before rejecting the null hypothesis, or larger sample sizes.
The non-parametric counterpart of both the t and z tests are: the Sign Test
*TABLE 1
Recommended methods of cooking HYPOTHESIS TESTING
MY CLAIM IS PARAMETRIC NONPARAMETRIC
ABOUT A: CLAIM ASSUMPTION TEST-/STATISTIC TEST/STATISTIC
mean Class A as a higher n EMBED Equation.DSMT4 30 or s.d known Z Sign Test IQ than average. n< 30, s.d. unknown T Wilcox/Mann WhitneyU
(U)
proportion 75% of voters prefer np> 5 and nq> 5 Z
candidate
standard This instrument normal population x2 .Kruskal-Wallis H
deviation has less errors than others (H)
two means EZ diet is more effective dependent T Sign Test than DF diet? independent T or Z U
The percent of cures is the
same for those using drug A
as Drug B. (A low t or U value would indicate that the proportions are similar. )
two standard The ages of group A are
deviations more homogeneous that the F (H)
ages of group B.
two proportions There are more Democrats Z Sign Test
in Chicago than LA.
relationship Smoking related to cancer. .Pearson r Spearman r
between 2 (If r is close to 0, then no relation)
variables
Are two variables F H
dependent?
How close do expected k variables x2
values agree with df = k-1
observed (aka Goodness of fit)
.ANOVA comparing 3 or more (compute: variances between F Kruskal-Wallis
means sample means/total variances ) (H)
df: num:=( k - 1) den = k(n-1);K= no. of groups n= amt in each group
Assumptions for ANOVA: normal distribution, equal variances from each sample. However, George E.P. Box demonstrated that as long as the sample sizes are equal (or near equal) the variances can be up to nine times as large and the results from ANOVA will continue to be essentially reliable. However, If the data don't fit these basic assumptions we can always use the non-Parametric version (Kruskal-Wallis).If a significant F ratio is found, another test can be employed to determine where the significance lies. One of these is Tukeys HSD (honestly significant difference).
Contingency table (two way ANOVA): A table of observed frequencies where the rows correspond to one variable and the columns another. It is used to see if 2 variables are dependent but can not be used to determine what the relationship is between the two variables.
STUDY OF 1000 DEATHS OF MALES
Cancer Heart Disease Other
Smoking 135 310 205
Non-Smoking 55 155 140
Note: The values in the table are observed values. These would be compared to expected values. A x2 statistics would be computed. A large x2 value indicates there is a relation between variables
We are now going to examine some felicitous applications for other statistical tests. You might wish to scan the list and see if you can identify similarities between the examples given and any of the hypotheses that you are planning to test.
TESTING CLAIMS ABOUT 2 MEANS
In this section we will discuss a claim made about two means (that the mean of one group is less than, greater than, or equal to the mean of another group). The researcher will first need to determine if the groups are dependent, i.e. the values in one sample are related to the values in another sample. This includes: before and after tests; tests involving spouses, relationships between an employer and an employee; or if the groups are independent, i.e. the values in one sample are not related to the values in another sample. This includes comparing an experimental group to a control group, or samples from two different populations like the eating habits of people in Michigan vs. Hawaii.
If the researcher can answer yes to one of the questions below, then the identical statistical test described in this section can be employed. Is the researcher claiming that:
___1. One product, program or treatment is better than another?
___2. One group is better (or worse ) than another? (with respect to some variable).
__ 3. An experimental program was effective?
Many real and practical situations involve testing hypotheses made about two population means. For example, a manufacturer may want to compare output on two different machines to see if they obtain the same result. A nutritionist might wish to compare the weight loss that results from patients on two different diet plans to determine which one is more effective. A psychologist may want to test for a difference in mean reaction times between men and women to determine if women respond quicker than men in an emergency situation.
If the two samples (groups) are dependent -- the values in one sample are related to the values in the other in some way, a t- statistic is computed and a simple paired t-test; may be used to test your claim. Computing the differences between the related means and then obtaining the mean of all these differences obtain this t-statistic.
If the two samples are independent i.e. the values in one sample are not related to the values in the other, and the size of each group, n, is 30 or more, or the standard deviations of the population, then a simple z statistics may be computed and a paired z-test could be ordered. In this case, the differences in the population means are computed and subtracted from the differences in the sample means. The result is divided by the square root of the sum of each variance divided by the respective sample size.
BUT if the two samples are: independent, the sample size (n) is less than 30 for each group, and the population standard deviation is not known then whom do you call? The F (team) test. The F test is used first to see if the standard deviations are equal. (A relatively small f value indicates that the standard deviations are the same). If the f-value is "relatively" small, then the researcher, or much more likely a computer, would need to perform a t-test to test the claim. This involves a very hackneyed computation. However, if the f-test was to yield a "relatively" large F value, this would lead to a more benign t-test .
Once the mean and standard deviation is computed for each sample, it is customary to identify the group with the larger standard deviation as group 1 and the other sample as group 2.
The non-parametric counterpart of the paired Z or T-test is the Wilcoxan signed-ranks test, if samples are dependent and the Wilcoxon Rank-Sum Test, if samples are independent
TESTING CLAIMS ABOUT 3 OR MORE MEANS
If the researcher can answer yes to one of the questions below, then the identical statistical test described in this section can be employed. Is there a claim that:
___1. There is a difference between three or more products, programs, or treatments?
___2. There is a different outcome from the same program, product or treatment among three or more different groups?
Claims about 3 or more means, require the creation of an F-statistic and the performing of an F test is on the menu. Here the researcher or most likely the computer, will compare the variances between the samples to the variances within the samples. The nickname for what's happening here is ANOVA; (Analysis of Variance). It is an extension of the t-test, which is used for testing two means. The null hypothesis is that the means are equal.
An F value close to 1 would indicate that there are no significant differences between the sample means. A "relatively large" F value will cause you to reject the null hypothesis and conclude that the means in the samples are not equal. Some important things to know about The F distribution:
1. It is not symmetric, it is skewed to the right.
2. The values of F can be 0 or positive, but not negative.
3. There is a different F distribution for each pair of degrees of freedom for the numerator and denominator.
Note: If you are asking: "Why are we dealing with variances when the claim is about means?" you would be asking a very good question. The answer is that the variance, or the standard deviation squared, is determined by, and dependent on, the mean, so it is actually all in the family!
FOR YOUR INFORMATION AND EDUCATION:
The method of ANOVA owes its beginning to Sir Ronald A. Fisher (1890-1962) and received its early impetus because of its applications to problems in agricultural research. It is such a powerful statistical tool that it has since found applications in just about every branch of scientific research, including economics, psychology, marketing research, and industrial engineering, to mention just a few.
The following example will be testing a claim about three means obtained from a sample Using the ssss candoall model. A report on the findings follows the example.
Example: A study was done to investigate the time in minutes for three police precincts to arrive at the scene of a crime. Sample results from similar types of crimes are:
A: 7 4 4 3
sample size: n = 4, mean. x = 4.5, variance, s2 = 3.0
B: 9 5 7
sample size: n = 3, mean x = 7.0, variance, s2= 4.2
C: 2 3 5 3 8
sample size: n = 5, mean x = 4.2, variance, s2= 5.7
At the EMBED Equation.DSMT4 = 0.05 significance level, test the claim that the precincts have the same mean reaction time to similar crimes.
What are the pre-test s's?
Assumptions for ANOVA: normal distribution, equal variances from each sample. However, George E.P. Box demonstrated that as long as the sample sizes are equal (or near equal) the variances can be up to nine times as large and the results from ANOVA will continue to be essentially reliable.
(a) What is the Substantive hypothesis;?
[What does the researcher think will happen?]
The reaction times are similar in the three precincts.
b) How large is the sample size that was studied?
The three groups have sample sizes 4,3, and 5 respectively.
c) What descriptive Statistic was determined by the sample?
The means and variances for each group were determined.
Now we are ready to take the information obtained in (a), (b), and (c) and employ the eight step CANDOALL recipe to test this hypothesis.
We will determine if the claim; The reaction times are similar is statistically correct.
1. Identify the Claim (C) to be tested and express it in symbolic form.
a = b = c
That is, there is a claim that the mean reaction time in each precinct is the same
2. Express in symbolic form the Alternative (A) statement that would be true if the original claim is false.
a EMBED Equation.DSMT4 b EMBED Equation.DSMT4 c
Remember we must cover all possibilities
3. Identify the Null (N) and alternative hypothesis.
Note: The null hypothesis should be the one that contains no change (an equal sign).
Ho: a = b = c (Null hypothesis)
H1: a EMBED Equation.DSMT4 b EMBED Equation.DSMT4 c (Alternative hypothesis)
Remember: A statistical test is designed to reject or fail to reject (accept) the statistical null hypothesis being examined.
4. Decide (D) on the level of significance, alpha ( EMBED Equation.DSMT4 ), based on the seriousness of a type I error
Note: This is the mistake of rejecting the null hypothesis when it is in fact true. Make alpha small if the consequences of rejecting a true alpha are severe. The smaller the alpha value, the less likely you will be to reject the null hypothesis. Alpha values 0.05 and 0.01 are very common.
EMBED Equation.DSMT4 = 0.05
5. Order (O) a statistical test and sampling distribution that is relevant to the study.* (see Table 1)
Since the claim involves data from three groups and we wish to test the hypothesis that the differences among the sample means is due to chance, we can use the ANOVA test.
Note: The following assumptions apply when using the ANOVA: The population has a normal distribution, the populations have the same variance (or standard deviation or similar sample sizes); the samples are random and independent of each other.
6. Perform the Arithmetic (A) and determine: the test statistic, the critical value and the critical region.
Note: It would be best for this to be performed on a computer.
To perform an ANOVA test we need to compute:
The number of samples, k,
k =3
The mean of all the times, X
x = 5.0
The variance between the samples - this is found by subtracting the mean (5.0) from the variance of each sample, squaring the differences, then multiplying each by the sample size and finally adding up the results for each sample.
The variance within the samples - this is found by multiplying the variance of each sample by one less than the number in the sample, adding the results, this equals 39.8, and then dividing by the total population minus the number of samples, 9.
The variance within the samples = 4.4222
The test statistic is F = variance between samples
variance within samples
F = 1.8317
The variance between the samples = 8.1
The degrees of freedom in the numerator = k-1 = 3-1 = 2. The degrees of freedom in the denominator = n-k = 12-3 = 9.
Note: Degrees of freedom are the number of values that are free to vary after certain restrictions have been imposed on all values. For example, if 10 scores must total 80, then we can freely assign values to the first 9 scores, but the tenth score would then be determined so that there would be 9 degrees of freedom. In a test using an F statistic we need to find the degrees of freedom in both the numerator and the denominator.
The critical value of F = 4.2565. (this can be found on a table or from a computer program).
7. Look (L) to reject or fail to reject the null hypothesis.
Note: This is a right-tailed test since the F statistic yields only positive values.
Since the test statistic of F = 1.8317 does not exceed the critical value of F = 4.2565, we fail to reject the null hypothesis that the means are equal.
Note: The shape of an F distribution is slightly different for each sample size n. The EMBED Equation.DSMT4 = 0.05 level employs us to find an F value that will separate the curve into 2 unequal regions; the smaller one with an area of 0.05 (5%), and the larger one with an area of 100%- 5% or 95% (0.95). (Often referred to as a 95% confidence level).
8. In Lay terms (L) write what happened.
There is not sufficient sample evidence to warrant rejection of the claim that the means are equal.
Note: In order for statistics to make sense in research it is important to use a rigorously controlled design in which other factors are forced to be constant. The design of the experiment is critically important, and no statistical calisthenics can salvage a poor design.
Writing About This Study in a Research Paper
If this study were to be published in a research journal, the following script could be used to summarize the statistical findings. This information usually appears in the data analysis section of a document but could also be properly placed in the section where the conclusion of the study is found, or even in the methodology section of the paper. This information would also be very appropriate to place in the abstract of the study.
A study was conducted to investigate the time in minutes for three police precincts to arrive at the scene of a crime. Sample results from similar types of crimes were found to be:
A: 7 4 4 3
sample size: n = 4, mean, x, = 4.5, variance, s2 = 3.0
B: 9 5 7
sample size: n = 3, mean, x = 7.0, variance, s2 = 4.2
C: 2 3 5 3 8
sample size: n = 5, mean, x = 4.2 variance, s2 = 5.7
At the EMBED Equation.DSMT4 =0.05 significance level, the claim that the precincts had the same mean reaction time to similar crimes was tested. The null hypothesis is the claim that the samples come from populations with the same mean:
Ho: a = b = c (Null hypothesis)
H1: a EMBED Equation.DSMT4 b EMBED Equation.DSMT4 c (Alternative hypothesis)
To determine if there are any statistically significant differences between the means of the three groups, an ANOVA test was performed. The groups were similar in size and the level of measurement was ratio data. An F distribution was employed to compare the two different estimates of the variance common to the different groups (i.e. variation between samples, and variation within the samples). A test statistic of F = 1.8317 was obtained. With 2 degrees of freedom for the numerator and 9 degrees of freedom for the denominator, the critical F value of 4.2565 was determined. Since the test F does not exceed the critical F value, the null hypothesis was not rejected. There is not sufficient sample evidence to reject the claim that the mean values were equal.
The non-parametric counterpart of ANOVA is: the Kruskal-Wallis Test
TESTING A CLAIM ABOUT PROPORTIONS/PERCENTAGES
If the answer to one of the questions below is yes, then the identical statistical test described in this section could be employed. Is the researcher claiming that:
___1. A certain percent or ratio is higher or lower than what is believed?
___2. There is a characteristic of a group that is actually prevalent in a higher or lower percent
Data at the nominal (name only) level of measurement lacks an real numerical significance and is essentially qualitative in nature. One way to make a quantitative analysis when qualitative data are obtained, is to represent that data in the form of a percentage or a ratio. This representation is very useful in a variety of applications, including surveys, polls, and quality control considerations involving the percentage of defective parts.
A Z- test; will work fine here provided that the size of the population is large enough. The condition is that:
EMBED Equation.DSMT4 and EMBED Equation.DSMT4
where, as always, n = sample size, p = population and q = 1- p
Note: The p in the test of proportions different than the "p- value we use to determine significance in hypothesis testing. It is important to be aware that in mathematics often times the same symbol can have more than one interpretation. While doing mathematics keep this in mind and remember to learn the meaning of a symbol in its context.
Example: If a manager believes that less than 48% of her employees support the companys dress code, the claim can be checked based on the response of a random sample of employees,
If 720 employees were sampled and 54.2% actually favored the dress code, then to check the managers claim, the researcher could perform a test of hypothesis to determine if the actual value of 0.542 is significantly different from the value of 0.48. Here, n =720, p = .48, q =0.542. The conditions EMBED Equation.DSMT4 and EMBED Equation.DSMT4 are met since 720(.48) = 345.6 and 720(.542) = 390.24. The z-value would be 3.33. This would lead us to reject the null hypothesis and conclude that this is probably a low estimate.
TESTING CLAIMS ABOUT STANDARD DEVIATIONS AND VARIABILITY
Many real and practical situations demand decisions or inferences about variances and standard deviations. In manufacturing, quality control engineers want to ensure that a product is on the average acceptable but also want to produce items of consistent quality so there are as few defects as possible. Consistency is measured by variances.
FOR YOUR INFORMATION AND EDUCATION:
During World War II, 35,000 American engineers and technicians were taught to use statistics to improve the quality of war material through the efforts of Dr. W. Edwards Deming (born in Sioux City, Iowa, on October 14, 1900). Deming's work was brought to the attention of the Union of Japanese Scientists and Engineers (JUSE). JUSE called upon Deming to help its members increase productivity. Deming convinced the Japanese people that quality drives profits up. The rebirth of Japanese industry and its worldwide success is attributed to the ideas and the teachings of Deming. In gratitude, the late Emperor Hirohito awarded Japans Second Order Medal of the Sacred Treasure Deming.
If you can answer yes to one of the questions below, you can use the identical statistical test described in this section. Are you claiming that:
___1. a product, program, or treatment has more or less variability than the standard?
___2. a product, program, or treatment is more or less consistent than the standard?
To test claims involving variability, the researcher usually turns to a Chi-square ( x2) statistic.
FOR YOUR INFORMATION AND EDUCATION:
Both the t and Chi square ( x2) distributions have a slightly different shape depending on n, the number in the sample. For this reason, the researcher needs to determine the "degrees of freedom" to find out what shape curve will be used to obtain the test statistics.
The "degrees of freedom" refer to the number of observations or scores minus the number of parameters that are being studied. (Informally, it is the number of times you can miss a certain targeted number and still have a chance of obtaining that desired outcome). When the researcher uses a sample size of (n) to investigate one parameter, e.g. a mean or standard deviation, the degrees of freedom equals n-1. When investigating a relationship between 2 variables then the degrees of freedom are: ( n-2). The test statistics used in tests of hypothesis involving variances or standard deviations, is chi-square, X 2
Example: A supermarket finds that the average check out waiting time for a customer on Saturday mornings is 8 minutes with a standard deviation of 6.2 minutes. One Saturday management experimented with a single queue. They sampled 25 customers and found that the average waiting time remained 8 minutes, but the standard deviation went down to 3.8 minutes.
To test the claim that the single line causes lower variation in waiting time, a computed chi-square value would be: X 2 = 9.016 and there would be 24 degrees of freedom since n = 25. The null hypothesis would be that the new line produced a standard deviation of waiting time greater than or equal to 6.2 This would yield a one-tail (left) test. The critical value would be 13.48, and we would reject the null hypothesis if the computed value were less than the critical value. Since 9.016 < 13.48 we would reject the null hypothesis and conclude that this method seems to lower the variation in waiting time.
TESTING A CLAIM ABOUT THE RELATION BETWEEN TWO VARIABLES (CORRELATION AND REGRESSION ANALYSIS)
Many real and practical situations demand decisions or inferences about how data from a certain variable can be used to determine the value of some other related variable.
For example, a Florida study of the number of powerboat registrations and the number of accidental manatee deaths confirmed that there was a significant positive correlation. As a result, Florida created coastal sanctuaries where powerboats are prohibited so that manatees could thrive.
A study in Sweden found that there was a higher incidence of leukemia among children who lived within 300 meters of a high-tension power line during a 25-year period. This led Sweden's government to consider regulations that would reduce housing in close proximity to high-tension power lines.
If you can answer yes to the questions below, the researcher can use the identical statistical test described in this section. Is the researcher claiming that:
___1. There is a relationship or correlation between two factors, two events or two characteristics and
__ 2. The data are at least of the interval measure.
In regression and correlation analysis the data are:
1. Record the information in table form.
2. A scatter diagram is usually created to see any "obvious" relationship or trends.
3. The correlation coefficient r (Rho) aka the Pearson Correlation Coefficient factor, to obtain objective analysis that will uncover the magnitude and significance of the relationship between the variables.
A test is performed to determine if r is statistically significant.
If r is statistically significant then regression analysis can be used to determine the relationship between the variables.
Example: Suppose randomly selected students are given a standard IQ test and then tested for their levels of math anxiety using a MARS test:
1. Record information in table form:
IQ(I) 103 113 119 107 78 153 111 128 135 86
MARS(M) 75 68 65 70 86 21 85 45 24 73
The researchers hypothesis is that students with higher IQ's have lower levels of math anxiety. [Note: The independent variable (x) is IQ scores, which are being used to predict the dependent variable (y) is MARS scores].
Ho: r = 0 (there is no relationship)
H1: r 0 (there is a relationship)
Note: These will usually be the hypotheses in regression analysis.
2. Draw a scatter diagram:
The points in the figure above seem to follow a downward pattern, so we might conclude that there is a relationship between IQ and levels of Math anxiety, but this is somewhat subjective.
3. Compute r
To obtain a more precise and objective analysis we can compute the linear coefficient constant, r. Computing r is a tedious exercise in arithmetic but practically any statistical computer program or scientific calculator would willingly help you along. In our example the very friendly program, STATVIEW was used to determine our r = -.882
Some of the properties or this number r are:
1. The computed value of r must be between (-1) and (+1). (If it's not then someone or something messed up.)
2. A strong positive correlation would yield an r-value close to (+1), a strong negative linear correlation would be close to (-1).
3. If r is close to 0 we conclude that there is no significant linear correlation between (X) and (Y).
Checking the table, we find that with a sample size of 10, (n = 10) , the value r = -.882 indicating a strong negative correlation between measures of IQ and measures of math anxiety levels. The r-squared number (.779) indicates that a persons IQ could explain 77.9% of a persons MARS score.
4. If there is a significant relation, then regression analysis is used to determine what that relationship is. If the relation is linear, the equation of the line of best fit can be determined. [For 2 variables, the equation of a line can be expressed as y = mx + b, where m is the slope and b is the y intercept]
Thus, the equation of the line of best fit would be
Y = -.914 I + 165.04
The non-parametric counterpart to r is the Spearman's rank correlation coefficient (rs) or r.
More on Correlational statistics
Warning: Correlation does not imply CAUSATION!
The Purpose of Correlational Research is to find Co-relationships between two or more variables with the hope of better understanding the conditions and events we encounter and with the hope of making predictions about the future. [From the Annals of Chaos Theory: Predictions are usually very difficult- especially if they are about the Future; Predictions are like diapers, both need to be changed often and for the same reason!].
As was noted previously, the linear correlation coefficient, r, measures the strength of the linear relationship between two paired variables in a sample. If there is a linear correlation, that is if r is "large enough," between two variables, then regression analysis is used to identify the relationship with the hope of predicting one variable from the other.
Note: If there is no significant linear correlation, then a regression equation cannot be used to make predictions.
A regression equation based on old data is not necessarily valid now. The regression equation relating used car prices and ages of cars is no longer usable if it is based on data from the 1960's. Often a scattergram is plotted to get a visual view of the correlation and possible regression equation.
Note: Nonlinear relationships can also be determined, but due to the fact that more complex mathematics are used to describe and interpret data, they are used considerably less often. The following are characteristics of all linear correlational studies:
1. Main research questions are stated as null hypotheses, i.e. "no" relationship exists between the variables being studied.
2. In simple Correlation, there are two measures for each individual in the sample.
3. To apply parametric methods, there must be at least 30 individuals in the study.
4. Can be used to measure "the degree" of relationships, not simply whether a relationship exists.
5. A perfect positive correlation is 1.00; a perfect negative (inverse) is -1.00.
6. A correlation of 0 indicates no linear relationship exists.
7. If when two variables x and y, are correlated so that r = .5, then we say that (0.5) 2 or 0.25 or 25% of their variation is common, or variable x can predict 25% of the variance in y.
Bivariate correlation is when there are only 2 variables being investigated. The following definitions help us determine which statistical test can be used to determine correlation and regression.
Continuous scores: Scores can be measured using a rational scale.
Ranked data: Likert Scale, Class Rankings
Dichotomy: subjects classified into two categories- Republican Vs. Democrat
Artificial - pass fail (arbitrary decision); true dichotomy ( male-female).
The Pearson Product Moment Correlation Coefficient (that is a mouthful!) or simply the Pearson r, is the most common measure of the strength of the linear relationship between two variables. It is named for Karl Pearson (1857-1936) who originally developed it. The Spearman Rank Correlation Coefficient, or Spearman r, (which we performed above), used for ranked data or when you have a sample size less than 30 (n<30), is the second most popular measure of the strength of the linear relationship between two variables. To measure of the strength of the linear relationship between test items for reliability purposes, the Cronbach alpha is the most efficient method of measuring the internal consistency. The following is a table to determine what statistical technique is best used with respect to the type of data the researcher collects.
Technique Symbol Variable 1 Variable 2 Remarks
Pearson r Continuous Continuous Smallest standard of error
Spearman Rank r Ranks Ranks Used also when n< 30
Kendall's tau t Ranks Ranks Used for n < 10
Biserial Correlation a/bis Artificial Continuous Sometimes exceeds 1
(Cronbach) Dichotomy often used in item
analysis.
Widespread biserial r/wbis Artificial Continuous Looking for extremes
correlation Dichotomy on Variable 1
Point- biserial r/pbis True Continuous Yields lower correlation correlation dichotomy than r/biserial
Tetrachoric correlation r/t Artificial Artificial Used when Var 1 and 2
Dichotomy Dichotomy can be split arbitrarily
(example: self-confidence Vs Internal Locus of Control)
Phi coefficient f True Dichotomy True Dichotomy
Correlation ratio eta h Continuous Continuous Nonlinear Relationships
Multivariate Correlational statistics
If you wish to test a claim that multiple independent variables might be used to make a prediction about a dependent variable, you have several possible tests that can be constructed. Such studies involve Multivariate Correlational statistics
Discriminate Analysis - Used to determine the correlation between two or more predictor variables and a dichotomous criterion variable. The main use of discriminant analysis is to predict group membership (e.g. success/non-success) from a set of predictors. If a set of variables is found which provide satisfactory discrimination, classification equations can be derived, their use checked out through hit/rate tables, and if good, they can be used to classify new subjects who were not in the original analysis. In order to use discriminate analysis certain assumptions (conditions) must be met:
* At least twice the number of subjects as variables in study.
* Each groups has at least n=# of variables.
* Groups have the same variance/covariance structures.
* All variables are normally distributed.
Canonical correlation - Used to predict a combination of several criterion variables from a combination of several predictor variables.
Path Analysis- Used to test theories about hypothesized causal links between variables that are correlated.
Factor Analysis - Used to reduce a large number of variables to a few factors by combining variables that are moderately or highly correlated with one another.
Differential analysis - Used to examine correlation between variables among homogeneous subgroups within a sample, can be used to identify moderator variables that improve a measure's predictive validity.
Multiple Regression - Used to determine the correlation between a criterion variable and a combination of two or more predictor variables. As in any regression method we need the following conditions to be met: We are investigating linear relationships; for each x value, y is a random variable having a normal distribution. All of the y variables have the same variance; for a given value of x, the distribution of y values has a mean that lies on the regression line.
Note: results are not seriously affected if departures from normal distributions and equal variances are not too extreme.
The following example illustrates how a researcher might use different multivariate correlational statistics in a research project.
Example: Suppose a researcher has, among other data, scores on three measures for a group of teachers working overseas:
1. years of experience as a teacher
2. extent of travel while growing up
3. tolerance for ambiguity.
Research Question: Can these measures (or other factors) predict the degree of adaptation to the overseas culture they are working on?
Discriminate Analysis : Hypothesis (1) : The outcome is dichotomous between those who adapted well and those who adapted poorly based on these three measures. Hypothesis (2): Knowing these three factors could be used to predict success.
Multiple Regression: Hypothesis: Some combination of the three predictor measures correlates better with predicting the outcome measure than any one predictor alone.
Canonical Correlation: Hypothesis: several measures of adaptation could be quantified, i.e. adaptation to food, climate, customs, etc. based on these predictors.
Path Analysis - Hypothesis: Childhood travel experience leads to tolerance for ambiguity and desire for travel as an adult, and this makes it more likely that a teacher will score high on these predictors which will lead them to seek an overseas teaching experience and adapt well to the experience.
Factor Analysis - Suppose there are 5 more ( a total of 8) adaptive measures which could be determined. All eight measures can be examined to determine whether they cluster into groups such as: education, experience, personality traits, etc.
ANALYSIS OF COVARIANCE
A "yes" on question one below will lead you to a different type of statistical test involving bivariate data. If the researcher is claiming that:
___1. Two groups being compared come from the same population and contain similar characteristics.
When subjects are divided into 2 groups (perhaps a control group and an experimental group), or if two different treatments on two different groups will be given, then problems in randomization and matching the groups might be a concern.
A statistical process called analysis of covariance (ANCOVA) has been developed to equate the groups on certain relevant variables identified prior to the investigation. Pre-test mean scores are often used as covariates.
The following guidelines should be used in the analysis of covariance:
1. The correlation of the covariate (some variable different than the one you are testing) and the response variable should be statistically and educationally significant.
2. The covariate should have a high reliability.
3. The covariate should be a variable that is measured prior to any portions of the treatments.
4. The conditions of homogeneity of regression should be met. The slopes of the regression lines describing the linear relationships of the criterion variable and the covariate from cell to cell must not be statistically different.
5. All follow-up procedures for significant interaction or "post-hoc comparisons" should be made with the adjusted cell or adjusted marginal means.
ANCOVA is a transformation from raw scores to adjusted scores which take into account the effects of the covariate. ANCOVA allows us to compensate somewhat when groups are selected by non random methods
The non-parametric counterpart of ANCOVA is: the Runs Test.
If your hypothesis is that many variables or factors are contributing to a certain condition, you may wish to use multiple regression analysis. This is similar to linear regression analysis with a significant increase in number crunching. If this is not how you wish to spend several hours of your day,.we recommend that you employ a computer to crank out the numerical information necessary to use multiple regression analysis.
CONTINGENCY TABLES:
Sometimes a researcher is only interested in the following:
___1. Whether or not two variables are dependent on one another, (e.g. are death and smoking dependent variables; are SAT scores and high school grades independent variables?)
To test this type of claim a contingency table could be used, with the null hypothesis being that the variables are independent. Setting up a contingency table is easy; the rows are one variable the columns another. In contingency table analysis (also called two-way ANOVA) the researcher determines how closely the amount in each cell coincides with the expected value of each cell if the two variables were independent.
The following contingency table lists the response to a bill pertaining to gun control.
In favor Opposed
Northeast 10 30
Southeast 15 25
Northwest 35 10
Southwest 10 25
Notice that cell 1 indicates that 10 people in the Northeast were in favor of the bill.
Example: In the previous contingency table, 40 out of 160 (1/4) of those surveyed were from the Northeast. If the two variables were independent, you would expect 1/2 of that amount (20) to be in favor of the amendment since there were only two choices. We would be checking to see if the observed value of 10 was significantly different from the expected value of 20.
To determine how close the expected values are to the actual values, the test statistic chi-square is determined. Small values of chi-square support the claim of independence between the two variables. That is, chi-square will be small when observed and expected frequencies are close. Large values of chi-square would cause the null hypothesis to be rejected and reflect significant differences between observed and expected frequencies.
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