What are the differences between a one-sided and a two-sided test?
Only use one-sided test when there is a clear reason for doing so. For example from economic theory or empirical evidence. Has more statistical pwer to detect an effect in one direction than a two-tailed test. Will occur when effects only can exist in one direction, or if the researchers only care about one direction (not recommended tho).
The difference lies in the alternative hypothesis. In one case you are testing if B1 is only greater or only lower than 0. In the other case, you are testing with the possibility of both scenarios.
Same nullhypothesis, different alternative hypothesis. Construction of the t-statistics is the same. Only difference is how you interpret the t-statistics
What is a two-tailed test and how do you perform it?
If we want to test if the mean is statistically and significantly equal to x, we can do a two-sided hypothesis. In other words, we want to test if B1 = 0. That gives us:
H0: B1 = 0
HA: B1 is not 0
Think of how normal distribution looks. If we use a significance level of 0.05 (or alpha = 0,05), the two tailed test will test the probability with an alpha of 0,025 on both tails.
- First compute the standard error of Y, which is an estimator of the standard deviation of the sampling distribution of Y.
- Compute the t-statistics
- Compute the p-value. P- value is the smallest significance level at which the null hypothesis could be rejected.
- Or use t-statistics: Reject H0 if the t-statistic is larger than absolute value of 1,96
Alternatively to the third step, you could compare the t-statistics to the critical value appropriate for the test with its significance level, that is the absolute value of 1,96 if you are testing on a 5%. Reject H0 if the t-statistic is larger than absolute value of 1,96.
What is a one-sided test and how to perform it?
In a one-sided test, the alternative hypothesis will be if B1 is either lower or if its higher than for example 0. A one-sided test should only be used when there is a clear reason of doing so. This reason can come from economic theory, your knowledge etc. You now test with an alpha of 0,05 on one tail. Not with 0,025 on each tail.
What is the p-value?
P-value is the smallest significance level at which the null hypothesis could be rejected.
Confidence interval for a regression coefficient
- A range where we are 95% certain that the real regression line is (if tested w 5% significance)
- Set of all values than cannot be rejected at 5% significance
- Interval that has 95% of containing the true value
When is it appropriate to do a Two-Sided Hypothesis?
- Testing hypothesis about the population mean
- Testing hypothesis about the slope B1
- Reporting regression equations and application to test scores
How to test when X is an Binary/Dummy variable
- D can only take two values, no “line” or “slope”
- Refer to B1 as the coefficient to D
- If B1 = 0, the Dummy will not be significant
- So divide B1 with its SER, same as normal
errors in statistical hypothesis tests
Type 1: Rejecting the null hypothesis when it is true
Type 2: Not rejecting the null hypothesis when it is false
significance level
The significance level is the probability of rejecting the null hypothesis when it is in fact true. a 5% significance level says that we have a prob of 5% of rejecting null when its true.
significance probability
The probability of drawing a statistic at least as adverse to the null hypothesis as the one you computed in your sample, assuming that the null hypothesis is true.
• What does a confidence interval tell you?
- true population is with 95% certainty within the confidence interval
• What is the problem of testing joint hypotheses with t-tests?
If we were to run t-tests on all and reject the whole regression if one turned
out significant, the size of the test would depend on the correlation between t1
and t2.
• What is meant by the size of a test?
In hypothesis testing, the size of a test is the probability of committing a Type I error, that is, of incorrectly rejecting the null hypothesis when it is true.
What is a p-value?
- P-value is the probability of rejecting the null hypothesis, when it should not be rejected (Type 1 error).
- We usually use 5% significance level, meaning that if a medicine is tested and it doesn’t have a real effect, the result will tell us it has an effect 1/20 times.
- The lower the p-value is, the greater statistics significance of our alternative hypothesis to be true
- A smaller p-value means that there is stronger evidence in favor of the alternative hypothesis
What do you need to conduct a hypothesis test?
- A hypothesis to be tested (usually described as the null hypothesis) and an alternative against which it can be tested.
- A test statistic where the distribution is known under the null hypothesis.
- We want to test or “prove” that our null hypothesis is correct” - A decision rule which tells us when to reject the null hypothesis and when not to reject it.
What is a critical value?
- A value that separates two regions:
1) Critical Region: Set of values that lead to Rejection of H0
2) Acceptance Region: Set of Values where H0 is not rejected
- Absolute > Critical = Reject H0
- Absolute < Critical = Keep H0
What is a P-value?
- P-value is the probability of rejecting the null hypothesis, when it should not be rejected (Type 1 error).
- We usually use 5% significance level, meaning that if a medicine is tested and it doesn’t have a real effect, the result will tell us it has an effect 1/20 times.
- The lower the p-value is, the greater statistics significance of our alternative hypothesis to be true
What are degrees of freedom?
- independent values that are free to vary in a data set
Intuitively:
A data set of four numbers. Three of the values are 4, 4, 4. and average of data is 4.
This must mean that the last number also has to be 4. It must be 4, it is not allowed to vary
What happen to a confidence interval when then sample is bigger and bigger?
It becomes smaller and smaller., sample size increase, more precise
