Question

1. What hypothesis is being tested to determine statistical significance of a coefficient?

2. Draw a t-distribution. Identify p-value on the graph for H0: B>=0 and Ha: B<0 hypothesis and t-statistics=0. Identify α=5% on the graph.

3. Draw a t-distribution. Identify p-value on the graph for H0: B<=0 and Ha: B>0 hypothesis and t-statistics=0. Identify α=5% on the graph.

4. Draw a t-distribution. Identify p-value on the graph for two-tail hypothesis and t-statistics=1.645. Identify α=5% on the graph.

Answer 1

1.The sample data are used to compute r, the correlation coefficient for the sample. If we had data for the entire population, we could find the population correlation coefficient. But because we have only have sample data, we cannot calculate the population correlation coefficient. The sample correlation coefficient, r, is our estimate of the unknown population correlation coefficient.

The symbol for the population correlation coefficient is ρ, the Greek letter “rho.”

ρ = population correlation coefficient (unknown)

r = sample correlation coefficient (known; calculated from sample data)

The hypothesis test lets us decide whether the value of the population correlation coefficient

ρ is “close to zero” or “significantly different from zero”. We decide this based on the sample correlation coefficient r and the sample size n.

2.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) , then it is "unlikely." And, if the P-value is large, say more than , then it is "likely."

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

3.The critical value approach involves determining "likely" or "unlikely" by determining whether or not the observed test statistic is more extreme than would be expected if the null hypothesis were true. That is, it entails comparing the observed test statistic to some cutoff value, called the "critical value." If the test statistic is more extreme than the critical value, then the null hypothesis is rejected in favor of the alternative hypothesis. If the test statistic is not as extreme as the critical value, then the null hypothesis is not rejected.

4.The research or alternative hypothesis can take one of three forms. An investigator might believe that the parameter has increased, decreased or changed. For example, an investigator might hypothesize:  

H1: μ > μ 0 , where μ0 is the comparator or null value (e.g., μ0 =191 in our example about weight in men in 2006) and an increase is hypothesized - this type of test is called an upper-tailed test;

H1: μ < μ0 , where a decrease is hypothesized and this is called a lower-tailed test; or

H1: μ ≠ μ 0, where a difference is hypothesized and this is called a two-tailed test.