[S]uppose we hypothesize that the true B1 takes a particular numerical value, e.g., B1 = 1. Our task now is to test this hypothesis.
In the language of hypothesis testing a hypothesis such as B1 = 1 is called the null hypothesis and is generally denoted by the symbol H0. Thus H0: B1 = 1. The null hypothesis is usually tested against an alternative hypothesis, denoted by the symbol H1. The alternative hypothesis can take one of three forms:
H1: B1 > 1, which is called a one-sided alternative hypothesis, or
H1: B1 < 1, also a one-sided alternative hypothesis, or
H1: B1 not equal 1, which is called a two-sided alternative hypothesis. That is the true value is either greater or less than 1.
The first thing we need to do to test our hypothesis is to calculate at t-Test statistic. The theory behind the statistic is beyond the scope of this article. Essentially what we are doing is calculating a statistic which can be tested against a t distribution to determine how probable it is that the true value of the coefficient is equal to some hypothesized value. When our hypothesis is B1 = 1 we denote our t-Statistic as t1(B1=1) and it can be calculated by the formula:
t1(B1=1) = (b1 - B1 / se1)
Lets try this for our intercept data. Recall we had the following data:
b1 = 0.47
se1 = 0.23
t1(B1=1) = (0.47 1) / 0.23 = 2.0435
So t1(B1=1) is 2.0435. We can also calculate our t-test for the hypothesis that the slope variable is equal to -0.4:
b2 = -0.31
se2 = 0.03
t2(B2= -0.4) = ((-0.31) (-0.4)) / 0.23 = 3.0000
So t2(B2= -0.4) is 3.0000. Next we have to convert these into p-values. The p-value "may be defined as the lowest significance level at which a null hypothesis can be rejected...As a rule, the smaller the p value, the stronger is the evidence against the null hypothesis." (Gujarati, 113) As a standard rule of thumb, if the p-value is lower than 0.05, we reject the null hypothesis and accept the alternative hypothesis. This means that if the p-value associated with the test t1(B1=1) is less than 0.05 we reject the hypothesis that B1=1 and accept the hypothesis that B1 not equal to 1. If the associated p-value is equal to or greater than 0.05, we do just the opposite, that is we accept the null hypothesis that B1=1.
Calculating the p-valueUnfortunately, you cannot calculate the p-value. To obtain a p-value, you generally have to look it up in a chart. Most standard statistics and econometrics books contain a p-value chart in the back of the book. Fortunately with the advent of the internet, theres a much simpler way of obtaining p-values. The site Graphpad Quickcalcs: One sample t test allows you to quickly and easily obtain p-values. Using this site, heres how you obtain a p-value for each test.
Steps Needed to Estimate a p-value for B1=1
- Click on the radio box containing Enter mean, SEM and N. Mean is the parameter value we estimated, SEM is the standard error, and N is the number of observations.
- Enter 0.47 in the box labelled Mean:.
- Enter 0.23 in the box labelled SEM:
- Enter 219 in the box labelled N:, as this is the number of observations we had.
- Under " 3. Specify the hypothetical mean value" click on the radio button beside the blank box. In that box enter 1, as that is our hypothesis.
- Click Calculate Now
P value and statistical significance:
The two-tailed P value equals 0.0221
By conventional criteria, this difference is considered to be statistically significant.