Five persons, who were suffering from depression, attended 10 one-hour counseling sessions. The following table gives the depression scores (on a scale of 1 to 10) of these five persons before and after attending the counseling sessions. Note that a higher score means that a person has a worse case of depression. Before After 7.5 4.3 6.4 5.1 7.0 6.2 9.2 6.9 7.8 7.2 Let the paired difference be the score before minus the score after attending the counseling sessions. The null hypothesis is that the mean of the population paired differences is equal to zero (i.e. Attending the counseling sessions does not change depression score.) The alternative hypothesis is that the mean of the population paired differences is not equal to zero (i.e., attending the counseling sessions does change the depression score). The significance level is 5%. 20. What is the mean of the sample paired differences, rounded to two decimal places? 1.92 21. What is the standard deviation of the paired differences, rounded to three decimal places? 22. What is the standard deviation of the mean of the sample paired differences, rounded to three decimal places? 23. Calculate the 95% confidence interval for the mean of the population paired differences that corresponds to these data, rounded to two decimal places. 24. What are the critical values of t for the hypothesis test? A) -2.776 and 2.776 C) -2.132 and 2.132 B) -2.571 and 2.571 D) -2.015 and 2.015 25. What is the value of the test statistic, t, rounded to three decimal places? 26. What is the p-value for this test, rounded to four decimal places? 27. Do you reject or fail to reject the null hypothesis? State your answer as "reject" or "fail to reject", but don't include the quotation marks.
Inverse Normal Distribution
The method used for finding the corresponding z-critical value in a normal distribution using the known probability is said to be an inverse normal distribution. The inverse normal distribution is a continuous probability distribution with a family of two parameters.
Mean, Median, Mode
It is a descriptive summary of a data set. It can be defined by using some of the measures. The central tendencies do not provide information regarding individual data from the dataset. However, they give a summary of the data set. The central tendency or measure of central tendency is a central or typical value for a probability distribution.
Z-Scores
A z-score is a unit of measurement used in statistics to describe the position of a raw score in terms of its distance from the mean, measured with reference to standard deviation from the mean. Z-scores are useful in statistics because they allow comparison between two scores that belong to different normal distributions.
Five persons, who were suffering from depression, attended 10 one-hour counseling sessions. The following table gives the depression scores (on a scale of 1 to 10) of these five persons before and after attending the counseling sessions. Note that a higher score means that a person has a worse case of depression.
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Before |
After |
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7.5 |
4.3 |
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6.4 |
5.1 |
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7.0 |
6.2 |
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9.2 |
6.9 |
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7.8 |
7.2 |
Let the paired difference be the score before minus the score after attending the counseling sessions. The null hypothesis is that the mean of the population paired differences is equal to zero (i.e. Attending the counseling sessions does not change depression score.) The alternative hypothesis is that the mean of the population paired differences is not equal to zero (i.e., attending the counseling sessions does change the depression score). The significance level is 5%.
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20. |
What is the mean of the sample paired differences, rounded to two decimal places? 1.92 |
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21. |
What is the standard deviation of the paired differences, rounded to three decimal places?
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22. |
What is the standard deviation of the mean of the sample paired differences, rounded to three decimal places? |
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23. |
Calculate the 95% confidence interval for the mean of the population paired differences that corresponds to these data, rounded to two decimal places.
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24. |
What are the critical values of t for the hypothesis test? |
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A) |
-2.776 and 2.776 |
C) |
-2.132 and 2.132 |
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B) |
-2.571 and 2.571 |
D) |
-2.015 and 2.015 |
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25. |
What is the value of the test statistic, t, rounded to three decimal places?
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26. |
What is the p-value for this test, rounded to four decimal places?
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27. |
Do you reject or fail to reject the null hypothesis? State your answer as "reject" or "fail to reject", but don't include the quotation marks.
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