04/21/23 Slide 1

SOLVING THE PROBLEM

A one-sample t-test of a population mean requires that the variable be quantitative.

A one-sample test of a population mean tests the null hypothesis that the population mean based on our sample data is not really different from some value specified for the population mean, e.g. identified in previous research.

The hypothesis is evaluated using probabilities for a sampling distribution that is normal (the t distribution). This requires the determination that our data supports the use of a normal distribution. We satisfy this requirement either •by demonstrating that our sample data which is representative of the population is reasonably normal (symmetric, unimodal, and without extreme outliers), •or by demonstrating that our sample is sufficiently large that the sampling distribution from which our sample comes will follow a normal distribution even when though our individual sample does not (the Central Limit Theorem).

If our sample mean is close to the specified population mean (using the standard error of the distribution as the measure of closeness), the probability that our sample could be drawn from a population with the specified population mean will be high, and we fail to reject the null hypothesis and do not interpret the results.

04/21/23 Slide 2

If our sample mean is far away from the specified population mean (using the standard error of the distribution as the measure of distance), the probability that our sample could be drawn from a population with the specified population mean will be low. We reject the null hypothesis and interpret the results of the test, consistent with the alternative hypothesis that explains the difference between the population mean based on our sample and the population mean specified for the test.

The decision that our sample mean is close or distant from the specified population mean is based on a comparison of the probability of the t-test statistic to the alpha level of significance established for the problem.

04/21/23 Slide 3

The introductory statement in the question indicates:• The data set to use (GSS2000R)• The variable to use in the analysis: total family

income [income98]• The task to accomplish (one-sample t-test of a

population mean)• The population mean (16.5)• The level of significance (0.05, two-tailed)

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The first statement asks about the level of measurement.

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"Total family income" [income98] is quantitative (ordinal treated as quantitative), satisfying the level of measurement requirement for a one-sample t-test of a population mean.

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A one-sample t-test of a population mean requires that the distribution of the variable satisfy the nearly normal condition, which we will operationally define as having skewness and kurtosis between -1.0 and +1.0, and having no outliers with standard scores equal to or smaller than -3.0 or equal to or larger than +3.0.

To justify the use of probabilities based on a normal sampling distribution in testing hypotheses, either the distribution of the variable must satisfy the nearly normal condition or the size of the sample must be sufficiently large to generate a normal sampling distribution under the Central Limit Theorem.

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To evaluate the variables conformity to the nearly normal condition, we will use descriptive statistics and standard scores.

To compute the descriptive statistics and standard scores, select the Descriptive Statistics > Descriptives command from the Analyze menu.

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Move the variable for the analysis income98 to the Variable(s) list box.

Click on the Options button to select optional statistics.

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The check boxes for Mean and Std. Deviation are already marked by default.

Click on Continue button to close the dialog box.

Mark the Kurtosis and Skewness check boxes. This will provide the statistics for assessing normality.

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Click on the OK button to produce the output.

Mark the check box Save standardized values as variables.

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"Total family income" [income98] satisfied the criteria for a normal distribution

The kurtosis of the distribution (-0.248) was between -1.0 and +1.0.

. The skewness of the distribution (-0.628) was between -1.0 and +1.0.

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There were no outliers that had a standard score less than or equal to -3.0.

Sort the column Zincome98 in ascending order to show any negative outliers at the top of the column.

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There were no outliers that had a standard score greater than or equal to +3.0.

Sort the column Zincome98 in descending order to show any positive outliers at the top of the column.

04/21/23 Slide 14

"Total family income" [income98] satisfied the criteria for a normal distribution. The skewness of the distribution (-0.628) was between -1.0 and +1.0 and the kurtosis of the distribution (-0.248) was between -1.0 and +1.0.

There were no outliers that had a standard score less than or equal to -3.0 or greater than or equal to +3.0.

Mark the statement as correct.

04/21/23 Slide 15

To apply the Central Limit Theorem for a one-sample t-test of a population mean requires that the sample have 40 or more cases.

Though we have satisfied the nearly normal condition and do not need to utilize the Central Limit Theorem to justify the use of probabilities based on the normal distribution, we will still examine the sample size.

04/21/23 Slide 16

The number of valid cases available for the test was 229, larger than requirement of 40 cases to apply the Central Limit Theorem.

However, since the distribution of "total family income" satisfied the nearly normal condition, we do not need to make use of the Central Limit Theorem to satisfy the sampling distribution requirements of a one-sample t-test of a population mean.

04/21/23 Slide 17

The number of valid cases available for the test was 229, larger than requirement of 40 cases to apply the Central Limit Theorem.

Mark the statement as correct.

04/21/23 Slide 18

The next statement asks us to identify the mean for the sample data and the standard error of the sampling distribution.

To answer this question, we need to produce the output for the one-sample t-test.

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To produce the one-sample t-test of a population mean, select the Compare Means > One-Sample T Test command from the Analyze menu.

04/21/23 Slide 20

First, move the variable income98 to the Test Variable(s) list box.

Third, click on the OK button to produce the output.

Second, type the value of the population mean that we are testing against, i.e. the mean based on previous research (16.5).

04/21/23 Slide 21

The standard error of the sampling distribution was correctly identified as 0.349.

The correct value for the sample mean was 15.7

The values we need are in the One-Sample Statistics table.

04/21/23 Slide 22

The correct value for the sample mean was 15.7. The standard error of the sampling distribution was correctly identified as 0.349.

Mark the check box as correct.

04/21/23 Slide 23

The next statement asks us about the null hypothesis for the one-sample t-test.

We should check to make certain the relationship is stated correctly, and that we verify the correct for the population mean that we are testing against.

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The null hypothesis for a one-sample t-test of a population mean states that the estimated population mean of income98 based on our data (15.7) is not different from the population mean of income98 based on previous research (16.5), i.e., the two estimates of the true population mean are equivalent.

Mark the statement as correct.

04/21/23 Slide 25

The next statement asks us to relate the t-test to the data in our problem.

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The t-test statistic is based on the difference between the estimated population mean based on our data and the hypothesized population mean based on previous research (15.668 - 16.500 = -0.832) relative to the standard error of the sampling distribution (0.349).

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The statement is correct and contains the correct values for both the difference in means and the sampling error that we would typically expect to find in the sampling distribution for income98.

Mark the statement as correct.

04/21/23 Slide 28

The next statement asks about the probability for the comparison made by the t-test. i.e. what is the probability that sample mean with the value for our data could be drawn from a population with the mean based on previous research and the standard error estimated from our data.

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The probability that a sample with a mean of 15.7 could be drawn from a population with a mean of 16.5 was p = .018, not p = .038.

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The probability that a sample with a mean of 15.7 could be drawn from a population with a mean of 16.5 was p = .018, not p = .038.

The statement is not correct and the check box is left empty.

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When the p-value for the statistical test is less than or equal to alpha, we reject the null hypothesis and interpret the results of the test. If the p-value is greater than alpha, we fail to reject the null hypothesis and do not interpret the result.

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The correct p-value for this test (p = .018) is less than or equal to the alpha level of significance (p = .050) supporting the conclusion to reject the null hypothesis.

Mark the statement as correct.

04/21/23 Slide 33

The final statement asks us to interpret the result of our statistical test as a finding in the context of the problem we created.

We only interpret the results when the null hypothesis is rejected.

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Since we rejected the null hypothesis and since the mean of "total family income" in our sample (15.7) is actually smaller than the mean reported in previous research (16.5), it is reasonable to suggest that true mean of "total family income" in the population has decreased.

Mark the check box as correct.

04/21/23

Variable is quantitative?

Yes

Do not mark check box.

Mark statement check box.

No

Mark only “None of the above.”

Stop.

04/21/23 Slide 37

Yes

Variable nearly normal distribution?

Do not mark check box.

No

Mark statement check box.

Nearly normal:•Skewness between -1.0 and +1.0•Kurtosis between -1.0 and +1.0•Z-scores between -3.0 and +3.0 CLT applicable

(Sample size ≥ 40)?

Do not mark check box.

Mark statement check box.

Stop.

CLT stands for Central Limit Theorem.

If the difference variable is not normal and the sample size is less than 40, the test is not appropriate.

Yes

No

CLT applicable(Sample size ≥ 40)?

Yes

No

Do not mark check box.

The path is complicated because we check two conditions, but only one needs to be correct to continue.

04/21/23 Slide 38

YesDo not mark

check box.Mark statement

check box.

Sample mean and standard error

correct?No

Yes

Do not mark check box.

No

Mark statement check box.

H0: mean = mean from previous research

04/21/23 Slide 39

T-test accurately described?

Yes

Do not mark check box.

Mark statement check box.

No

P-value (sig.) stated correctly?

Yes

Do not mark check box.

Mark statement check box.

No

04/21/23 Slide 40

Interpretation is stated correctly?

Yes

Do not mark check box.

Mark statement check box.

No

Yes

Do not mark check box.

No

Mark statement check box.

Reject H0 is correct decision (p ≤ alpha)?

Stop.

We interpret results only if we reject null hypothesis.

04/21/23 Slide 41

To use the T-Tests script, select the type of T-Test desired by marking one of the option buttons.

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Select a test variable from the list.

Specify the value for the population mean to test against.

Click on the OK button to produce the output.

04/21/23 Slide 43

The script produces skewness and kurtosis as descriptive statistics that can be used to evaluate the near normal condition.

The script also creates a table of outliers that will list any outliers with a z-score less than or equal to -3.0 or greater than or equal to + 3.0.

If there are no outliers, it will list the minimum and maximum z-score for the data.

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The T-Test output shows the number of cases for evaluating the applicability of the Central Limit Theorem.

The T-Test output provides the evidence needed to make a decision about the null hypothesis.

04/21/23 Slide 45

The script also produces a histogram and boxplot. that can be used to evaluate the distribution or confirm your interpretation of the statistical evidence.

To facilitate interpretation, the histogram can be overlaid with a dotplot showing more detail for the distribution of cases, lines showing the location of the mean and standard deviation units, and a normal curve.