Question

1. A researcher uses a simple linear regression to measure the relationship between the monthly salary (Salary measured in dollars) of data scientists and the number of years since being awarded a Master degree (Master Degree). A random sample of 80 observations was collected for the analysis. A researcher used the econometric model which has the following specification Salary,-β0 + β, Master-Degree, + εί, where i = 1, , 80 The (incomplete) Excel output of equation (1) is presented below SUMMARY OUTPUT Regression Statistics R-Square Observations ANOVA XxX 80 df MS Regression Residual Total XxX 126408.2 XxX 20514650.04 Standard Coefficients error t-stat Intercept Master_Degree 6589.12 150.95 136.14 13.52

Simple Linear regression

Answer 1

The regression model being estimated is

$\text{Salary}_i=\beta_0+\beta_1\text{Master\_Degree}_i+\epsilon_i$

where Salary is the Monthly salary (in Dollars)

Master_Degree is the number of years since being awarded a master degree.

a) The error term is $\epsilon_i\stackrel{iid}\sim \mathcal{N}(0,\sigma^2)$

The assumptions are

• The error term is i.i.d, that is the errors are independent and identically distributed.
• The error terms are normally distributed with mean 0 and constant variance

b) The estimated values of the coefficients are

The estimated value of intercept is $\hat{\beta}_0=6589.12$

The estimated value of the slope is $\hat{\beta}_1=150.95$

Ans: The estimated equation is

$\widehat{\text{Salary}}=6589.12+150.95\times \text{Master\_Degree}$

c) The estimated value of the slope coefficient is 150.95. The positive value indicates that number of years since being awarded a master degree and the monthly salary move in the same direction. That is the monthly salary would increase with the increase in number of years since being awarded a master degree. For each year increase in the number of years since being awarded a master degree, the monthly salary increases by $150.95.

It is reasonable that monthly salary would increase with the increase in number of years since being awarded a master degree, provided the person is gainfully employed (as a data scientist?) during these years in the field of interest. That is there is a corresponding increase in the number of years of experience working as a data scientist.

d) 90% confidence interval indicates that a significance level of $\alpha=1-90/100=0.10$ .

The critical t value is obtained using $P(T>t_{\alpha/2})=\alpha/2=0.1/2=0.05$ .

The number of observations is n=80. The degrees of freedom for t is n-2=80-2=78.

Using the t tables we can get the critical value of t for degrees of freedom df=60 as $t_{\alpha/2}=1.671$ and for df=120 as 1.658.

the value for df=78 will be something between these 2 values. We can either interpolate of use excel to get the exact value.

Using =T.INV.2T(0.1,78) we get a t value of 1.665

$t_{\alpha/2}=1.665$

We know the standard error of the slope estimate from the output

$s.e(\hat{\beta}_1)=13.52$

The 90% confidence interval is

$\begin{align*} &\hat{\beta}_1\pm t_{\alpha/2}s.e(\hat{\beta}_1)\\ \implies &150.95\pm 1.665\times 13.52\\ \implies &[128.44,173.46] \end{align*}$

ans: 90% confidence interval for the slope is [128.44,173.46]

e) There would be a positive relationship between monthly salary and number of years, if the slope coefficient is positive (that is >0).

That is we want to test the following hypotheses

$\begin{align*} &H_0:\beta_1=0\leftarrow\text{null hypothesis: There is no positive relationship between salary and number of years}\\ &H_a:\beta_1> 0\leftarrow\text{alternative hypothesis: There is a positive relationship between salary and number of years}\\ &\alpha=0.05\leftarrow\text{level of significance to test the hypotheses} \end{align*}$

The hypothesized value of the slope is $\begin{align*} \beta_{1H_0}=0 \end{align*}$

The test statistics is

$\begin{align*}t=\frac{\hat{\beta}_1- \beta_{1H_0}}{s.e(\hat{\beta}_1)}=\frac{150.95-0}{13.52}=11.165 \end{align*}$

this is a 1 tailed (right tailed) test (The alternative hypothesis has ">")

the p-value is P(T>11.165).

Using the excel function, =T.DIST.RT(11.165,78) we get p-value=3.83E-18

We will reject the null hypothesis if the p-value is less than the significance level.

Here the p-value of 0.000 is less than the significance level 0.05.

Hence we reject the null hypothesis.

We conclude that there is a significant positive relationship between monthly salary and number of years.

f) Using the ANOVA table

df MS Regression Residual Total 126408.2 20514650.04

the degrees of freedom for Residuals is df=n-2=80-2=78

The Mean square residuals is MSE=126408.2.

The sum of square residuals is

$\begin{align*}&\text{MSE}=\frac{\text{SSE}}{df}\\ \implies &\text{SSE}=\text{MSE}\times df\\ \implies &\text{SSE}=126408.2\times 78=9859839.6 \end{align*}$

Sum of square Total is SST=20514650.04

The coefficient of determination is

$\begin{align*}R^2=1-\frac{\text{SSE}}{\text{SST}}=\frac{9859839.6}{20514650.04}=0.5194 \end{align*}$

The value of coefficient of determination is 0.5194. It indicates that 51.94% of variation is monthly salary is explained by the variation is the number of years since being awarded a master degree.

g) The expected value of salary for master_degree=20 is

$\widehat{\text{Salary}}=6589.12+150.95\times 20=9608.12$

The expected monthly salary of a data scientist who has been awarded with a master degree for 20 years is $9,608.12