© 2008 McGraw-Hill Higher Education The Statistical Imagination Chapter 10. Hypothesis Testing II:...

© 2008 McGraw-Hill Higher Education

The Statistical Imagination

• Chapter 10.

Hypothesis Testing II:

Single-Sample Hypothesis Tests: Establishing the Representativeness of Samples


Single-Sample Hypothesis Tests

• Used to answer the question:

“Is a parameter equal to some chosen target value?”


Sources of Target Values

1. A known population parameter from a comparison group

2. Known parameters from a past time period

3. A statistical ideal

4. A population is sampled, and the sample statistics are compared to known population parameters to determine whether the sample is representative of the population


Small Single-SampleMeans Test

• Useful for testing a hypothesis that the mean of a population is equal to a target value


When to Use a Small Single Sample Means Test

1.There is only one variable

2. The level of measurement of the variable is interval/ratio

3. There is a single sample and population

4. n < 121 cases

5. There is a target value of the variable to which we may compare the mean of the sample


The Student’s t Sampling Distribution

• The sampling distribution curve used with especially small samples

(n < 121 cases) is called the

Student’s t-distribution

• The t-distribution is an approximately normal distribution


The t-distribution of Means

• For a sampling distribution of means, when the sample size is 121 or fewer cases, the probability curve begins to flatten into

a t-distribution• See Figure 10-1 in the text


Features of the t-distribution

• Standardized scores for the t-distribution are called t-scores and are computed like Z-scores

• The t-distribution, like the Z-distribution of the normal curve, allows us to calculate probabilities

• The t-distribution is symmetrical with mean, median, and mode centered, but the curve is flatter reflecting greater sampling error


The t-distribution Table

• The t-distribution table (Appendix B, Statistical Table C) is organized differently from the normal curve table

• See Table 10-1 in the text• The t-distribution table provides t-scores

for only the critical probabilities of .05, .01, and .001; that is, this table provides “critical t-scores”


Degrees of Freedom

• Degrees of freedom ( df ) are the number of opportunities in sampling to compensate for limitations, distortions, and potential weaknesses in statistical procedures

• Use of the t-distribution table requires the calculation of degrees of freedom

• For single-sample means tests, df = n - 1


More on Degrees of Freedom

• From repeated sampling we know that any single sample is only an estimate. An estimate can be distorted by limitations of the statistical procedures used to obtain it

• E.g., the mean is influenced by outliers• The larger the sample, the greater the opportunity

for an outlier to be neutralized in such a way as to not distort a sample mean

• A small sample is especially vulnerable to outliers


Z-Tests and t-Tests

• The small sample t-test applies when the sample size, n < 121; we use the t-distribution table

• However, regardless of sample size, all single-sample means tests are typically referred to as t-tests because the Z-test is a special case of a t-test, a t-test where n > 121; the t-distribution table includes critical scores for this situation under df = ∞


Features of a Single-Sample Means Test

• Follow the six steps of statistical inference

• Step 1: In general, state the H0 :

Parameter (sampled population) = known target parameter

• That is, the mean of a population

= a target value

• As in all hypotheses tests, the H0 and HA refer to population parameters


Features of a Single-Sample Means Test (cont.)

• Step 2. t-distribution with df = n - 1• The standard error is estimated by

dividing the sample standard deviation by the square root of the sample size, n

• Draw the t-distribution curve• Obtain the critical t-score from the t-table in Appendix B


The Critical Test Score for a t-Distribution

• Step 3: For a hypothesis test using the t-distribution curve, the critical test score is the specific t-score that is large enough to cause us to reject the H0

• After selecting the critical value from the t-distribution table, identify its location on the probability curve in Step 2 and estimate what kind of sample outcomes reach the critical region


Features of a Single-Sample Means Test (Continued)

• Step 4. The effect is the difference between the observed sample mean and the hypothesized target value

• The test statistic is the effect divided by the standard error

• The p-value is estimated from the t-distribution table


Relationships Among Mathematical Parts of the Test

• There are rules that assist in understanding the relationships among hypothesized parameters, observed sample statistics, the test statistic, the p-value, and the level of significance and its critical test score


Rule 1: The Test Effect and the Critical Test Score

• The H0 is rejected when the test effect is large enough that the test statistic is greater than the critical test score value

• If │tobserved │ > │ tα │, then p < α; reject H0


Rule 2: The Effect and the Size of the p-Value

• The larger the test effect and the test statistic, the smaller the p-value

• A large test effect implies a significant difference between what was hypothesized and what was observed

• A large test effect makes the H0 suspect


Rule 3: Rejection and Test Direction

• It is easier to reject with a one-tailed test than with a two-tailed test

• Choosing to use a one-tailed test must be done on the basis of theory or logic and is done in Step 1 of the six steps

• Do not look to Step 4, the sample outcome, to make a choice of test direction


Rule 4: The Level of Significance and Rejection

• The lower the level of significance, the harder it is to reject the H0

• The H0 is rejected when the p-value is less than or equal to α. The p-value must be especially small to be smaller than a small α


Rule 5: Fail to Reject When Findings are in the Wrong Direction

• When using a one-tailed test but the observed sample statistic falls in the opposite direction, immediately fail to reject the H0

• Even if found statistically significant, one cannot justify a wrong direction finding


Testing for Sample Representativeness

• Single-sample hypothesis tests are especially useful for determining whether a sample is representative of the population from which it was drawn

• To establish the representativeness of a sample, data must be acquired on some known parameters of the population to provide the target value for the test


Testing for Sample Representativeness (cont.)

• Single-sample hypothesis tests determine whether the test effect is (a) due to random sampling error or (b) due to a failure to sample and represent some aspect of the population adequately



• The H0 is stated as “the sample is representative”

• If the difference between the known parameter and the sample statistic is not statistically significant but simply due to sampling error, then we will not reject this H0



• The HA is stated as “the sample is not representative”

• If the difference between the known parameter and the sample statistic is significantly different, then we reject the H0 and conclude that the sampled population and targeted population are not one and the same – that there is sample bias



• Most tests of sample representativeness use nominal/ordinal variables, such as gender and race, because known parameters are available in government documents (e.g., the U.S. Census)

• These parameters provide the target values for such tests


Large Single-Sample Proportions Test

• The large single-sample proportions test is used for testing sample representativeness with nominal/ordinal variables


When to Use a Large Single-Sample Proportions Test

1. There is only one variable

2. It is a nominal/ordinal variable with

P = p [of success category] and Q = p [of failure category/categories]

3. There is a single sample and population

4. The sample size is sufficiently large that (psmaller) (n) > 5

5. There is a target value (known parameter) with which to compare the sample proportion


Features of a Large Single-Sample Proportions Test

• Step 1. Stating the H0:

Parameter (sampled population) = a known target parameter

• That is: Pu = a known target value


Features of a Large Single-Sample Proportions Test (cont.)

• Step 2. The sampling distribution of proportions is the approximately normal t-distribution with a standard error estimated by taking the square root of Pu times Qu divided by n

• In the t-distribution table, look to df = ∞; this is essentially a normal curve test



• Step 4. The effect is the difference between the observed sample proportion (Ps) and the hypothesized target value (Pu)

• The test statistic is the effect divided by the standard error

• The p-value is estimated from the t-distribution table



• Steps 5 and 6: When the H0 is rejected, focus on the interpretation of the HA, which is that the sample is NOT representative


Confidence Interval of the Mean When n is Small (n < 121 cases)

• For a confidence interval of the mean (Chapter 8), when the sample size is 121 or fewer cases, substitute the appropriate t-score in place of a Z-score

• tα in place of Zα


Statistical Follies: Sample Size and Tests of Representativeness

• When testing for sample representativeness, we wish to fail to reject the of H0 of no significant difference

• With a small sample and its large standard error, it is too easy to accomplish this, but there is the chance of Type II error, in this case, concluding the sample is representative when it is not


Statistical Follies: Sample Size and Tests of Representativeness (cont.)

• To avoid a Type II error, the sample size must be sufficiently large

• Statistical power is a test statistic’s probability of not incurring a Type II error for a given level of confidence

© 2008 McGraw-Hill Higher Education The Statistical Imagination Chapter 10. Hypothesis Testing II:...

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