SOLVING THE PROBLEM A one-sample t-test of a population mean requires that the variable be quantitative |
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1 | SOLVING THE PROBLEM A one-sample | 20 | Third, click on the OK button to |
t-test of a population mean requires that | produce the output. First, move the | ||
the variable be quantitative. A one-sample | variable income98 to the Test Variable(s) | ||
test of a population mean tests the null | list box. Second, type the value of the | ||
hypothesis that the population mean based | population mean that we are testing | ||
on our sample data is not really different | against, i.e. the mean based on previous | ||
from some value specified for the | research (16.5). 8/7/2015. Slide 20. | ||
population mean, e.g. identified in | 21 | The values we need are in the | |
previous research. The hypothesis is | One-Sample Statistics table. The correct | ||
evaluated using probabilities for a | value for the sample mean was 15.7. The | ||
sampling distribution that is normal (the | standard error of the sampling | ||
t distribution). This requires the | distribution was correctly identified as | ||
determination that our data supports the | 0.349. 8/7/2015. Slide 21. | ||
use of a normal distribution. We satisfy | 22 | The correct value for the sample mean | |
this requirement either by demonstrating | was 15.7. The standard error of the | ||
that our sample data which is | sampling distribution was correctly | ||
representative of the population is | identified as 0.349. Mark the check box as | ||
reasonably normal (symmetric, unimodal, | correct. 8/7/2015. Slide 22. | ||
and without extreme outliers), or by | 23 | The next statement asks us about the | |
demonstrating that our sample is | null hypothesis for the one-sample t-test. | ||
sufficiently large that the sampling | We should check to make certain the | ||
distribution from which our sample comes | relationship is stated correctly, and that | ||
will follow a normal distribution even | we verify the correct for the population | ||
when though our individual sample does not | mean that we are testing against. | ||
(the Central Limit Theorem). If our sample | 8/7/2015. Slide 23. | ||
mean is close to the specified population | 24 | The null hypothesis for a one-sample | |
mean (using the standard error of the | t-test of a population mean states that | ||
distribution as the measure of closeness), | the estimated population mean of income98 | ||
the probability that our sample could be | based on our data (15.7) is not different | ||
drawn from a population with the specified | from the population mean of income98 based | ||
population mean will be high, and we fail | on previous research (16.5), i.e., the two | ||
to reject the null hypothesis and do not | estimates of the true population mean are | ||
interpret the results. 8/7/2015. Slide 1. | equivalent. Mark the statement as correct. | ||
2 | If our sample mean is far away from | 8/7/2015. Slide 24. | |
the specified population mean (using the | 25 | The next statement asks us to relate | |
standard error of the distribution as the | the t-test to the data in our problem. | ||
measure of distance), the probability that | 8/7/2015. Slide 25. | ||
our sample could be drawn from a | 26 | The t-test statistic is based on the | |
population with the specified population | difference between the estimated | ||
mean will be low. We reject the null | population mean based on our data and the | ||
hypothesis and interpret the results of | hypothesized population mean based on | ||
the test, consistent with the alternative | previous research (15.668 - 16.500 = | ||
hypothesis that explains the difference | -0.832) relative to the standard error of | ||
between the population mean based on our | the sampling distribution (0.349). | ||
sample and the population mean specified | 8/7/2015. Slide 26. | ||
for the test. The decision that our sample | 27 | The statement is correct and contains | |
mean is close or distant from the | the correct values for both the difference | ||
specified population mean is based on a | in means and the sampling error that we | ||
comparison of the probability of the | would typically expect to find in the | ||
t-test statistic to the alpha level of | sampling distribution for income98. Mark | ||
significance established for the problem. | the statement as correct. 8/7/2015. Slide | ||
8/7/2015. Slide 2. | 27. | ||
3 | The introductory statement in the | 28 | The next statement asks about the |
question indicates: The data set to use | probability for the comparison made by the | ||
(GSS2000R) The variable to use in the | t-test. i.e. what is the probability that | ||
analysis: total family income [income98] | sample mean with the value for our data | ||
The task to accomplish (one-sample t-test | could be drawn from a population with the | ||
of a population mean) The population mean | mean based on previous research and the | ||
(16.5) The level of significance (0.05, | standard error estimated from our data. | ||
two-tailed). 8/7/2015. Slide 3. | 8/7/2015. Slide 28. | ||
4 | The first statement asks about the | 29 | The probability that a sample with a |
level of measurement. 8/7/2015. Slide 4. | mean of 15.7 could be drawn from a | ||
5 | "Total family income" | population with a mean of 16.5 was p = | |
[income98] is quantitative (ordinal | .018, not p = .038. 8/7/2015. Slide 29. | ||
treated as quantitative), satisfying the | 30 | The probability that a sample with a | |
level of measurement requirement for a | mean of 15.7 could be drawn from a | ||
one-sample t-test of a population mean. | population with a mean of 16.5 was p = | ||
8/7/2015. Slide 5. | .018, not p = .038. The statement is not | ||
6 | To justify the use of probabilities | correct and the check box is left empty. | |
based on a normal sampling distribution in | 8/7/2015. Slide 30. | ||
testing hypotheses, either the | 31 | When the p-value for the statistical | |
distribution of the variable must satisfy | test is less than or equal to alpha, we | ||
the nearly normal condition or the size of | reject the null hypothesis and interpret | ||
the sample must be sufficiently large to | the results of the test. If the p-value is | ||
generate a normal sampling distribution | greater than alpha, we fail to reject the | ||
under the Central Limit Theorem. A | null hypothesis and do not interpret the | ||
one-sample t-test of a population mean | result. 8/7/2015. Slide 31. | ||
requires that the distribution of the | 32 | The correct p-value for this test (p = | |
variable satisfy the nearly normal | .018) is less than or equal to the alpha | ||
condition, which we will operationally | level of significance (p = .050) | ||
define as having skewness and kurtosis | supporting the conclusion to reject the | ||
between -1.0 and +1.0, and having no | null hypothesis. Mark the statement as | ||
outliers with standard scores equal to or | correct. 8/7/2015. Slide 32. | ||
smaller than -3.0 or equal to or larger | 33 | The final statement asks us to | |
than +3.0. 8/7/2015. Slide 6. | interpret the result of our statistical | ||
7 | To evaluate the variables conformity | test as a finding in the context of the | |
to the nearly normal condition, we will | problem we created. We only interpret the | ||
use descriptive statistics and standard | results when the null hypothesis is | ||
scores. To compute the descriptive | rejected. 8/7/2015. Slide 33. | ||
statistics and standard scores, select the | 34 | 8/7/2015. Slide 34. | |
Descriptive Statistics > Descriptives | 35 | Since we rejected the null hypothesis | |
command from the Analyze menu. 8/7/2015. | and since the mean of "total family | ||
Slide 7. | income" in our sample (15.7) is | ||
8 | Move the variable for the analysis | actually smaller than the mean reported in | |
income98 to the Variable(s) list box. | previous research (16.5), it is reasonable | ||
Click on the Options button to select | to suggest that true mean of "total | ||
optional statistics. 8/7/2015. Slide 8. | family income" in the population has | ||
9 | The check boxes for Mean and Std. | decreased. Mark the check box as correct. | |
Deviation are already marked by default. | 8/7/2015. Slide 35. | ||
Click on Continue button to close the | 36 | Variable is quantitative? Do not mark | |
dialog box. Mark the Kurtosis and Skewness | check box. No. Mark only “None of the | ||
check boxes. This will provide the | above.”. Stop. Mark statement check box. | ||
statistics for assessing normality. | 8/7/2015. | ||
8/7/2015. Slide 9. | 37 | Variable nearly normal distribution? | |
10 | Click on the OK button to produce the | Nearly normal: Skewness between -1.0 and | |
output. Mark the check box Save | +1.0 Kurtosis between -1.0 and +1.0 | ||
standardized values as variables. | Z-scores between -3.0 and +3.0. Do not | ||
8/7/2015. Slide 10. | mark check box. No. CLT stands for Central | ||
11 | "Total family income" | Limit Theorem. CLT applicable (Sample size | |
[income98] satisfied the criteria for a | ? 40)? Mark statement check box. Do not | ||
normal distribution. . The skewness of the | mark check box. CLT applicable (Sample | ||
distribution (-0.628) was between -1.0 and | size ? 40)? Stop. If the difference | ||
+1.0. The kurtosis of the distribution | variable is not normal and the sample size | ||
(-0.248) was between -1.0 and +1.0. | is less than 40, the test is not | ||
8/7/2015. Slide 11. | appropriate. Do not mark check box. Mark | ||
12 | Sort the column Zincome98 in ascending | statement check box. The path is | |
order to show any negative outliers at the | complicated because we check two | ||
top of the column. There were no outliers | conditions, but only one needs to be | ||
that had a standard score less than or | correct to continue. 8/7/2015. Slide 37. | ||
equal to -3.0. 8/7/2015. Slide 12. | 38 | Sample mean and standard error | |
13 | Sort the column Zincome98 in | correct? No. Do not mark check box. Mark | |
descending order to show any positive | statement check box. H0: mean = mean from | ||
outliers at the top of the column. There | previous research. Do not mark check box. | ||
were no outliers that had a standard score | No. Mark statement check box. 8/7/2015. | ||
greater than or equal to +3.0. 8/7/2015. | Slide 38. | ||
Slide 13. | 39 | T-test accurately described? Do not | |
14 | "Total family income" | mark check box. No. Mark statement check | |
[income98] satisfied the criteria for a | box. P-value (sig.) stated correctly? Do | ||
normal distribution. The skewness of the | not mark check box. No. Mark statement | ||
distribution (-0.628) was between -1.0 and | check box. 8/7/2015. Slide 39. | ||
+1.0 and the kurtosis of the distribution | 40 | Reject H0 is correct decision (p ? | |
(-0.248) was between -1.0 and +1.0. There | alpha)? Do not mark check box. No. Stop. | ||
were no outliers that had a standard score | We interpret results only if we reject | ||
less than or equal to -3.0 or greater than | null hypothesis. Mark statement check box. | ||
or equal to +3.0. Mark the statement as | Interpretation is stated correctly? Do not | ||
correct. 8/7/2015. Slide 14. | mark check box. No. Mark statement check | ||
15 | Though we have satisfied the nearly | box. 8/7/2015. Slide 40. | |
normal condition and do not need to | 41 | To use the T-Tests script, select the | |
utilize the Central Limit Theorem to | type of T-Test desired by marking one of | ||
justify the use of probabilities based on | the option buttons. 8/7/2015. Slide 41. | ||
the normal distribution, we will still | 42 | Select a test variable from the list. | |
examine the sample size. To apply the | Specify the value for the population mean | ||
Central Limit Theorem for a one-sample | to test against. Click on the OK button to | ||
t-test of a population mean requires that | produce the output. 8/7/2015. Slide 42. | ||
the sample have 40 or more cases. | 43 | The script produces skewness and | |
8/7/2015. Slide 15. | kurtosis as descriptive statistics that | ||
16 | The number of valid cases available | can be used to evaluate the near normal | |
for the test was 229, larger than | condition. The script also creates a table | ||
requirement of 40 cases to apply the | of outliers that will list any outliers | ||
Central Limit Theorem. However, since the | with a z-score less than or equal to -3.0 | ||
distribution of "total family | or greater than or equal to + 3.0. If | ||
income" satisfied the nearly normal | there are no outliers, it will list the | ||
condition, we do not need to make use of | minimum and maximum z-score for the data. | ||
the Central Limit Theorem to satisfy the | 8/7/2015. Slide 43. | ||
sampling distribution requirements of a | 44 | The T-Test output shows the number of | |
one-sample t-test of a population mean. | cases for evaluating the applicability of | ||
8/7/2015. Slide 16. | the Central Limit Theorem. The T-Test | ||
17 | The number of valid cases available | output provides the evidence needed to | |
for the test was 229, larger than | make a decision about the null hypothesis. | ||
requirement of 40 cases to apply the | 8/7/2015. Slide 44. | ||
Central Limit Theorem. Mark the statement | 45 | The script also produces a histogram | |
as correct. 8/7/2015. Slide 17. | and boxplot. that can be used to evaluate | ||
18 | The next statement asks us to identify | the distribution or confirm your | |
the mean for the sample data and the | interpretation of the statistical | ||
standard error of the sampling | evidence. To facilitate interpretation, | ||
distribution. To answer this question, we | the histogram can be overlaid with a | ||
need to produce the output for the | dotplot showing more detail for the | ||
one-sample t-test. 8/7/2015. Slide 18. | distribution of cases, lines showing the | ||
19 | To produce the one-sample t-test of a | location of the mean and standard | |
population mean, select the Compare Means | deviation units, and a normal curve. | ||
> One-Sample T Test command from the | 8/7/2015. Slide 45. | ||
Analyze menu. 8/7/2015. Slide 19. | |||
SOLVING THE PROBLEM A one-sample t-test of a population mean requires that the variable be quantitative.ppt |
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