Question

Simple regression was employed to establish the effects of childhood exposure to lead. The effective sample size was about 122 subjects. The independent variable was the level of dentin lead (parts per million). Below are regressions using various dependent variables.

(a) Calculate the t statistic for each slope. From the p-values, which slopes differ from zero at α = .01? (Round your answers to 2 decimal places. Negative values should be indicated by a minus sign.)

Dependent Variable	R2			Estimated Slope			Std Error			tcalculated			p-value				Differ from 0?
Highest grade achieved		0.050			-0.029			0.010						.004			(Click to select)  No  Yes
Reading grade equivalent		0.126			-0.093			0.022						.000			(Click to select)  Yes  No
Class standing		0.078			-0.005			0.002						.014			(Click to select)  Yes  No
Absence from school		0.073			4.700			1.620						.004			(Click to select)  No  Yes
Grammatical reasoning		0.090			0.132			0.070						.062			(Click to select)  No  Yes
Vocabulary		0.119			-0.196			0.002						.000			(Click to select)  Yes  No
Hand-eye coordination		0.035			0.060			0.027						.028			(Click to select)  Yes  No
Reaction time		0.016			9.300			3.560						.010			(Click to select)  No  Yes
Minor antisocial behavior		0.037			-0.819			0.579						.160			(Click to select)  No  Yes

(b) It would be inappropriate to assume cause and effect without a better understanding of how the study was conducted.

• Yes

• No

Answer 1

The detailed calculation of the tstat and pvalue is shown using excel formula.

Also, the significance is indicated for each variable.

the explanation for determining the significance.

For each beta coefficient, we test the following hypothesis.

We check the pvalue associated with that variable,
if the pvalue is less than 0.01(level of significance specified), then we reject the null hypothesis and conclude that the variable is significant or is statistically different from zero. Hence it is a significant predictor of y.

if the pvalue is greater than 0.01(level of significance specified), then we fail to reject the null hypothesis and conclude that the variable is not significant or is not statistically different from zero. Hence it is not a significant predictor of y and can be dropped from the regression.

(b) It would be inappropriate to assume cause and effect without a better understanding of how the study was conducted.
Yes

Explanation:

Based on the correlation of the independent variable with the target variable, we cannot conclude a cause and effect relationship.

Understanding causality
Causality defines a cause and effect between to variables. In other words, it indicates a relationship between two variables (events) where one variable effects the other.

For example: Variable 1: Average number of hours spent preparing for exams
Variable 2: Score or marks in the exams.

Here we have a cause and effect relationship if you do not study you will very poor marks. But you study every day for 3 hours you will get a good score in the exams.

Differentiating correlation from causation

Correlation between two variables defines the strength and the direction of the linear relationship between two variable.
By strength we mean, how strong or weak is the association between the two variables.

The correlation coefficient takes a value between 0 and 1 and it can have a positive or negative sign depending on the relationship.

Higher the value, stronger is the relationship.
A positive sign indicates that as one variables increase or decreases, the other variable also increases or decreases in the same proportion.

A negative sign indicates that as one variable increases the other decreases and vice versa.

For causation, we need to have an empirical relationship between the variable.
But for correlation, we only examine the values and do check for any intrinsic connection between the variables.

The correlation between lung cancer and drinking alcohol. But we cannot conclude that drinking causes lung cancer. There could be many other reasons for lung cancer.

On the other have smoking and lung cancer can have a causal relationship and also a high correlation.

Hence correlation does not indicate causation, correlation only indicates the strength of the relationship, causation can be established only with empirical evidence.

Therefor with out understanding how the study was conducted assuming a cause and effect relationship is incorrect.