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Hypothesis Testing Using One-Sample t-Tests

Hypothesis Testing Using One-Sample t-Tests

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First we’ll consider our hypothesis that the intercept variable equals one. The idea behind this is explained quite well in Gujarati’s Essentials of Econometrics. On page 105 Gujarati describes hypothesis testing:

    “[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.”

In the above I’ve substituted in our hypothesis for Gujarati’s to make it easier to follow. In our case we want a two-sided alternative hypothesis, as we’re interested in knowing if B1 is equal to 1 or not equal to 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)

Let’s try this for our intercept data. Recall we had the following data:

Intercept

    b1 = 0.47
    se1 = 0.23
Our t-Statistic for the hypothesis that B1 = 1 is simply:

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:

X Variable

    b2 = -0.31
    se2 = 0.03
Our t-Statistic for the hypothesis that B2 = -0.4 is simply:

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-value

Unfortunately, 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, there’s 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, here’s 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”
You should get an output page. On the top of the output page you should see the following information:

    P value and statistical significance:
    The two-tailed P value equals 0.0221
    By conventional criteria, this difference is considered to be statistically significant.
So our p-value is 0.0221 which is less than 0.05. In this case we reject our null hypothesis and accept our alternative hypothesis. In our words, for this parameter, our theory did not match the data.

Be Sure to Continue to Page 3 of "Hypothesis Testing Using One-Sample t-Tests".

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