Hypothesis Testing Question and Answers 2024 final
exam
• We have no theoretical basis to specify a direction, but we do believe there is some difference
between the population mean and a specified value.
• H1 : μy ≠ some value
• H0 (The null hypothesis) is no difference.
• H0 : μy = some value
One-Tailed Tests
H1 is directional, specifying that the population mean is greater than or less than some specified value.
• If we state that the population mean is greater than some specified value, it is a right-tailed test. => H1
: μy > some value
• If we state that the population mean is less than some specified value, it is a left-tailed test. => H1 : μy
< some value
Non-directional hypotheses (two-tailed)
• Null hypothesis
• American workers work the same number of hours per week than workers in industrialized countries.
=> H0 : μy = 40 hrs
• Research hypothesis
• American workers do not work the same number of hours per week than workers in industrialized
countries. => H : μ ≠ 40 hrs
Directional hypotheses (one-tailed)
• Null hypothesis
,• American workers work the same number of hours per week than workers in industrialized countries.
=> H0 : μy = 40 hrs
• Research hypotheses
• American workers work more hours per week than workers in industrialized countries (right-tailed
test) => H : μ > 40 hrs
• American workers work less hours per week than workers in industrialized countries (left-tailed test)
=> H1 : μy < 40 hrs
**When in doubt, use a non-directional
hypothesis. **
Testing the Null Hypothesis
Rather than testing H1 directly we test the null hypothesis H0 (there is no real difference in the number
of hours worked) because we hope to reject the null hypothesis in order to provide support for the
research hypothesis.This is why H1 is also referred to as the "alternative hypothesis".
If we reject the H0 , we conclude that there is a real difference
Z statistic
• The obtained Z is the number of standard errors that our sample is from Y , assuming the null
hypothesis is true.
• What is the probability of finding this Z statistic?
• The probability corresponds to the area beyond Z. In this example, the area is on the right tail of the
curve.
• p < 0.0001
P value
•The probability of getting a sample mean of 43 hours if H0 is true is less than 0.0001.
•If there is no difference in the sample of American workers (43 hours) from the population mean (40
hours), then this sample mean would be extremely unlikely (less than 1 out of 1000 sample means).
, α and P value
•P value is the probability associated with the obtained test statistic.
•It is a measure of how unusual or rare our obtained statistic is.
•Since we want to reject H0, we need small p values.
•We also need to define a cut-off point. If p falls below this cut-off point, we reject the null hypothesis.
• α is our cut-off point, or level of significance at which the null hypothesis is rejected.
•α represents the level of risk we are willing to take in rejecting H0 if H0 is true.
•α is set in advance by the researcher.
Rejection Level and Decision-Rules
The decision to reject the null hypothesis is based on the alpha level.
Alpha is usually set at the 0.05, 0.01, or 0.001 level.
Alpha of 0.05 means that there is a 5 in 100 probability of observing a test statistic of that size due to
chance.
We are confident that 95% of the time we can reject the null hypothesis without making an error.
If the obtained probability (P) ≤ alpha , reject HO (null hypothesis)
Our differences are statistically significant.
If the obtained probability (P) > alpha , fail to reject HO
Our differences are not statistically significant.
The relationship between the p-value and alpha allow us to make statements regarding significance of
the results.
American works and the general population.
-The difference is statistically significant. It's not just due to random chance alone.
•If there is no difference in the sample of American workers (43 hours) from the population mean (40
hours), then this sample mean would be extremely unlikely (less than 1 out of 1000 sample means).
-Thus, we can reject this explanation.
-There really is a difference in the mean number of hours American works and the general population.
-The difference is statistically significant. It's not just due to random chance alone.
exam
• We have no theoretical basis to specify a direction, but we do believe there is some difference
between the population mean and a specified value.
• H1 : μy ≠ some value
• H0 (The null hypothesis) is no difference.
• H0 : μy = some value
One-Tailed Tests
H1 is directional, specifying that the population mean is greater than or less than some specified value.
• If we state that the population mean is greater than some specified value, it is a right-tailed test. => H1
: μy > some value
• If we state that the population mean is less than some specified value, it is a left-tailed test. => H1 : μy
< some value
Non-directional hypotheses (two-tailed)
• Null hypothesis
• American workers work the same number of hours per week than workers in industrialized countries.
=> H0 : μy = 40 hrs
• Research hypothesis
• American workers do not work the same number of hours per week than workers in industrialized
countries. => H : μ ≠ 40 hrs
Directional hypotheses (one-tailed)
• Null hypothesis
,• American workers work the same number of hours per week than workers in industrialized countries.
=> H0 : μy = 40 hrs
• Research hypotheses
• American workers work more hours per week than workers in industrialized countries (right-tailed
test) => H : μ > 40 hrs
• American workers work less hours per week than workers in industrialized countries (left-tailed test)
=> H1 : μy < 40 hrs
**When in doubt, use a non-directional
hypothesis. **
Testing the Null Hypothesis
Rather than testing H1 directly we test the null hypothesis H0 (there is no real difference in the number
of hours worked) because we hope to reject the null hypothesis in order to provide support for the
research hypothesis.This is why H1 is also referred to as the "alternative hypothesis".
If we reject the H0 , we conclude that there is a real difference
Z statistic
• The obtained Z is the number of standard errors that our sample is from Y , assuming the null
hypothesis is true.
• What is the probability of finding this Z statistic?
• The probability corresponds to the area beyond Z. In this example, the area is on the right tail of the
curve.
• p < 0.0001
P value
•The probability of getting a sample mean of 43 hours if H0 is true is less than 0.0001.
•If there is no difference in the sample of American workers (43 hours) from the population mean (40
hours), then this sample mean would be extremely unlikely (less than 1 out of 1000 sample means).
, α and P value
•P value is the probability associated with the obtained test statistic.
•It is a measure of how unusual or rare our obtained statistic is.
•Since we want to reject H0, we need small p values.
•We also need to define a cut-off point. If p falls below this cut-off point, we reject the null hypothesis.
• α is our cut-off point, or level of significance at which the null hypothesis is rejected.
•α represents the level of risk we are willing to take in rejecting H0 if H0 is true.
•α is set in advance by the researcher.
Rejection Level and Decision-Rules
The decision to reject the null hypothesis is based on the alpha level.
Alpha is usually set at the 0.05, 0.01, or 0.001 level.
Alpha of 0.05 means that there is a 5 in 100 probability of observing a test statistic of that size due to
chance.
We are confident that 95% of the time we can reject the null hypothesis without making an error.
If the obtained probability (P) ≤ alpha , reject HO (null hypothesis)
Our differences are statistically significant.
If the obtained probability (P) > alpha , fail to reject HO
Our differences are not statistically significant.
The relationship between the p-value and alpha allow us to make statements regarding significance of
the results.
American works and the general population.
-The difference is statistically significant. It's not just due to random chance alone.
•If there is no difference in the sample of American workers (43 hours) from the population mean (40
hours), then this sample mean would be extremely unlikely (less than 1 out of 1000 sample means).
-Thus, we can reject this explanation.
-There really is a difference in the mean number of hours American works and the general population.
-The difference is statistically significant. It's not just due to random chance alone.
