Question

If the sample size is small, the estimation results might be misleading. Why?

Explain the diferences between Statistically significant VS economically important.

How to use p-value in making a rejection decision?

How to interpret the 95% CI for the hypothesis test of H_0: β_j=0 against H_1: β_j≠0?

True or false: “If (β_j ) ̂ is outside CI, then we reject the null hypothesis.”

How can we test H_0: β_1=β_2 against H_1: β_1≠β_2?

Answer 1

(1) If the sample size is small, the estimation results might be misleading. Why?

Since smaller samples yield smaller power, a small sample size may not be able to detect an important difference. If there is strong evidence that the power of a procedure will indeed detect a difference of practical importance, then accepting the null hypothesis may be appropriate¹; otherwise it is not -- all we can legitimately say then is that we fail to reject the null hypothesis.

(2) Explain the diferences between Statistically significant VS economically important.

-Statistical Significance: We will look at the t-tests or p-values to determine whether or not to reject the null hypothesis (which says that the parameter is equal to 0) at a certain level of significance.

+ Statistical significance can be driven from a large estimate or a small standard error (which may result from a large sample size, meaning there are more variance in x variables)

+ A lack of statistical significance may be driven from small sample size or multicollinearity(meaning that there are correlations between x variables)

-Economic significance: we will look at the magnitude and the sign of the estimated coefficient. If the number turns out to be so small, that x variable does not really affect y.

In short, a coefficient is

statistically significant when it is quite precisely estimated,

and

economically significant when it is important.

(3) How to use p-value in making a rejection decision?

The P-value approach involves determining "likely" or "unlikely" by determining the probability — assuming the null hypothesis were true — of observing a more extreme test statistic in the direction of the alternative hypothesis than the one observed. If the P-value is small, say less than (or equal to) αα, then it is "unlikely." And, if the P-value is large, say more than αα, then it is "likely."

If the P-value is less than (or equal to) αα, then the null hypothesis is rejected in favor of the alternative hypothesis. And, if the P-value is greater than αα, then the null hypothesis is not rejected.

Specifically, the four steps involved in using the P-value approach to conducting any hypothesis test are:

1. Specify the null and alternative hypotheses.
2. Using the sample data and assuming the null hypothesis is true, calculate the value of the test statistic. Again, to conduct the hypothesis test for the population mean μ, we use the t-statistic t∗=(x¯−μ)/s/√n which follows a t-distribution with n - 1 degrees of freedom.
3. Using the known distribution of the test statistic, calculate the P-value: "If the null hypothesis is true, what is the probability that we'd observe a more extreme test statistic in the direction of the alternative hypothesis than we did?" (Note how this question is equivalent to the question answered in criminal trials: "If the defendant is innocent, what is the chance that we'd observe such extreme criminal evidence?")
4. Set the significance level, αα, the probability of making a Type I error to be small — 0.01, 0.05, or 0.10. Compare the P-value to αα. If the P-value is less than (or equal to) αα, reject the null hypothesis in favor of the alternative hypothesis. If the P-value is greater than αα, do not reject the null hypothesis.

(5)True or false: “If (β_j ) ̂ is outside CI, then we reject the null hypothesis.

TRUE

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