# Evaluate continuous variables.

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Consider the model Y = B_0 + B_1X_1+B_2X_2+e, e ~ N(0, σ^2) where the 'xs are all continuous variables. You also have data on a qualitative variable denoted d1 for each observation, where d1 = 1 if the observation belongs to group 1, and zero otherwise.

1) Assume you want to set an empirical model to see the expected value of Y was different for members of group 1. What model, null, and alternative hypothesis would you need to estimate and test, assuming identical continuous variable values? Show your work and explain. (Hint: use dummy variable)

2) What model, null, and alternative hypothesis would you need to estimate and test to allow for differences in both the intercept and the slope coefficient on X2? Show your work and explain. (Hint: you need an interaction term).

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#### Solution Preview

See the attached file.

1. To test whether group 1 has an effect or no, consider the following model.

Y = β0 + β1x1 + β2x2 + β3x3 + e.

All terms are same as the original model, x3 is a dummy variable where x3 = 1 if the observation falls into group 1 and x3 = 0 otherwise.

What want to see if Y is different if the observation is in group 1 (i.e. x3 = 1) or not in group 1 (i.e. x3 = 0). That is, we want to see if β3 is significantly not zero.

H0: β3 = 0 (i.e. being in group 1 or not has ...

#### Solution Summary

The solution assists with evaluating continuous variables.

Grading and Evaluation

A professor wants to conduct a study to know whether the grades she gives to her students affect their evaluations of her. She teaches four classes of ten students each. She has a theory that the grades and her evaluations are affected by the following characteristics of students:

A student's gender

A student's age

The hours of sleep a student gets per night

A student's year in the college

A student's current grade

The number of pets a student owns

Identify the independent and dependent variables in this study. For the independent variables, explain what can be the possible levels (subcategories within each variable) of the variables or the range of values.

Explain the relationship between the sample and the population in the study.

Identify the scales of measurement (nominal, ordinal, interval, and ratio) for each of the variables. Explain why the scale of measurement you chose is appropriate for this study. Some variables have more than one possible scale of measurement; so support your reasoning with examples.

In one of the professor's classes, students' evaluations of her (scored on a scale of 1 to 10) are as follows: 3, 5, 5, 5, 5, 6, 6, 8, 8, and 9. For these evaluations:

Calculate (ΣX)/n.

The university changes its scoring policy, and now the students' evaluations are scored on a scale of 5 to 15. For these evaluations:

Calculate (Σ(X + 5))/n.

The professor wants to see what would happen if each student gives her one point higher. Describe the formula to calculate that.

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