Hypothesis testing t-test

### An agricultural researcher has prior information about growing tomatoes – Growing tomatoes requires warm weather. – Each one-degree increase in temperature increases the amount of tomatoes growing in a standard bed by 3.1 kg. ### The researcher suspects that the above information is incorrect – He wants to conduct a test that will eventually verify or disprove the information. – He prepared 10 beds for an experiment under laboratory conditions, with different heating for each bed. – At the end of the experiment, he weighed the tomatoes from each bed. ### The econometric formulation of the model – The index i represents the observation number (bed number). – Represents the amount of tomatoes harvested at the end of the growing period in bed i. – Represents the temperature provided to the tomatoes in bed i. – The meaning of is the additional amount of tomatoes that will grow in each bed following a one-degree increase in temperature. ### Hypotheses – **Null hypothesis**: The prior information is correct. – **Alternative hypothesis**: The prior information is incorrect. ### Results of the experiment – The researcher obtained that and . – The result obtained (3) is 1.178 standard deviations below the prior information (3.1). ### Is this a large or small distance? – If the t-statistic (in absolute value) is less than 2, the distance is small and the estimation result is close to the prior information. – If the t-statistic (in absolute value) is greater than 2, the distance is large and the estimation result is far from the prior information. ### Conclusion – If the estimation result is close to the prior information, the prior information cannot be contradicted, and the null hypothesis is accepted. – If the estimation result is far from the prior information, the null hypothesis is rejected and the alternative hypothesis is accepted. ### In conclusion – The estimated result is close to the prior information: the null hypothesis is not rejected. – The estimated result is far from the prior information: the null hypothesis is rejected. ### Another example – The sample has 10 observations and the model has 2 parameters, so the number of degrees of freedom is 8.
– The value in the table corresponding to 8 degrees of freedom is 2.306.
– Since, we reject , and decide that the prior information is incorrect. ### Implicit Hypothesis – If we are presented with the following hypothesis to test: , we first handle the expression using algebra. – We calculate the t-statistic, compare it to 2 (or a more precise value from the t-table), and make a decision. ### Estimator Significance Test – A researcher estimates a model and obtains the results: ,
– Is the estimator significant?
– The question “Is the estimator significant?” translates to the following:
– (The estimator is not significant)
– (The estimator is significant) ### Another example – A researcher estimated the model using a sample of 22 observations.
– received that and .
– Calculate the t-statistic:
– The sample has 22 observations and the model has 2 parameters, so the number of degrees of freedom is 20.
– The corresponding value in the t table is 2.086.
– Since -, we reject the null hypothesis, and decide that the estimate is significant.
– Therefore, there is no need to change the model.