# Testing a hypothesis about the size of coefficients in a linear relationship

In the project, I was asked to determine whether

$0.5(\text{Father's Height} + \text{Mother's Height}) +6.5$

correctly predicts the height for boys.

To do this, I've collected 50 male students' height and their parents'. $n=50$

However, I am not sure how to run my hypothesis test. I used confidence interval to classify Tall, Average and Short with 90 percent of the sample are classified as average height.(Actually I'm not sure about the percentage too, is 90% a good choice?) Here the problem comes.

Now, I have 2 tests in my mind that can be used, chi-square goodness-of-fit test, and binomial test. By using chi-squared test, I'm going to test whether the formula correctly predicts the category a person falls into(eg. Short,Average).

If I use binomial test, I am going to test whether the probability of success prediction, $p\geq0.85$, and a prediction is considered as success if the predicted height differs from the actual height within 2.5cm. But, I don't know which are better. Or, can anyone suggest a better test?

You can also convert the equation you have into one where the regression will give you a direct test of the coefficients.

Intercept: if you subtract your hypothesised intercept (6.5) from your $Y$ variable (child height) and then run a regression the test whether the intercept is zero in the new model is the test that it has the hypothesised intercept with the unshifted $Y$.

Slope: you need to fit a model including the $X$ variable with a known coefficient (0.5). This is called an offset. Use $0.5 \times{} X$ as the offset and also include $X$ in the model as a covariate. The test of whether the coefficient of $X$ is zero in the new model is the same as the test of whether it is 0.5 in the model without offset.

Of course you would do both simultaneously.

You want to know whether there is a linear relationship between a boy's height and the combined height of their parents. More specifically, you have a theory about what the slope and intercept is in this linear relationship.

One way of approaching the problem would seem to be to set those predictions up as your null hypothesis, and then try to reject it.

You could fit the linear regression to this height data and then inspect the coefficients and their standard errors. If the predicted values (i.e. 0.5 and 6.5) do not sit within the standard errors (or CIs) then you can reject the null, and this would provide evidence that your hypothesised relationship between parent and child height is incorrect.