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Predicting CPI For a random sample of 100 students in a German university, the result of regressing the college cumulative performance index (CPI) on the high school grade (HSG), the average monthly attendance percentage (AMAP) and the average daily study time (ADST) follows. $$ \begin{aligned} &\text { Regression Equation }\\\ &\mathrm{CPI}=1.1362+0.6615 \mathrm{HSG}+0.2301 \mathrm{AMAP}+0.0075 \mathrm{ADST}\\\ &\begin{array}{l} \text { Coefficients } \\ \text { Term } & \text { Coef } & \text { SE Coef } & \text { T-Value } & \text { P-Value } \\ \text { Constant } & 1.1362 & 0.5731 & 1.98 & 0.050 \\ \text { HSG } & 0.6615 & 0.1578 & 4.19 & 0.000 \\ \text { AMAP } & 0.3301 & 0.1473 & 2.24 & 0.027 \\ \text { ADST } & 0.0075 & 0.0151 & 0.50 & 0.621 \\ \text { Model Summary } & & \\ & S & \mathrm{R}-\mathrm{sq} & & & \\ 0.218624 & 75.82 \% & & & \end{array} \end{aligned} $$ a. Explain in nontechnical terms what it means if the population slope coefficient for high school grade (HSG) equals 0 . b. Show all the steps for testing the hypothesis that this slope equals \(0 .\)

Step by step solution

01

Understand the Slope Coefficient

In regression analysis, the slope coefficient represents the change in the dependent variable (CPI) per unit change in the independent variable (HSG). Thus, if the slope coefficient for HSG equals 0, it implies that changes in HSG do not affect CPI. In other words, the High School Grade would have no influence on the College Performance Index if its slope coefficient is 0.

02

Set Up the Null and Alternative Hypotheses

To test whether the population slope coefficient for HSG is 0, we need to establish our null and alternative hypotheses. The null hypothesis (H_0): \(\beta_{HSG} = 0\), and the alternative hypothesis (H_a): \beta_{HSG} eq 0. These hypotheses mean we want to check if HSG does, in fact, have no effect on CPI (null) or does have an effect (alternative).

03

Gather Relevant Data

We need the coefficient, standard error, T-value, and P-value for HSG to test the hypothesis. From the data provided, the coefficient for HSG is 0.6615, the standard error (SE Coef) is 0.1578, the T-Value is 4.19, and the P-Value is 0.000.

04

Calculate and Interpret the T-Statistic

The T-statistic is calculated as the coefficient divided by its standard error. This is shown as \[T = \frac{0.6615}{0.1578} = 4.19.\]This calculated T-value is given, which implies that the test statistic calculated supports a strong effect of HSG on CPI.

05

Decision Rule Based on P-Value

Typically, we compare the P-value with a significance level (e.g., \(\alpha = 0.05\)). If the P-value is less than \(\alpha\), we reject the null hypothesis. In this case, the P-value is 0.000, which is much less than 0.05, indicating that the null hypothesis can be rejected.

06

Conclusion about the Hypothesis Test

With a P-value of 0.000, we have strong evidence to reject the null hypothesis that the HSG slope coefficient is 0. Thus, we conclude that there is a statistically significant relationship between High School Grades and the College Performance Index.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hypothesis Testing

Hypothesis testing is a statistical method that allows us to make decisions based on data. It is used to determine if there is enough evidence to reject a conjecture about a population parameter.
In the context of regression analysis, we often test if certain predictors significantly influence a dependent variable. Here, we test if the slope coefficient of High School Grade (HSG) affects the College Performance Index (CPI).
For this, we set up two hypotheses:

• The null hypothesis (\(H_0\)): \(\beta_{HSG} = 0\), meaning HSG does not impact CPI.
• The alternative hypothesis (\(H_a\)): \(\beta_{HSG} eq 0\), indicating HSG does influence CPI.

Hypothesis testing helps us determine if the result from our sample is due to chance or if it's significant enough to apply to a whole population. We use data like the coefficient, standard error, and P-value to make these determinations.

Slope Coefficient

The slope coefficient in a regression equation quantifies the relationship between an independent variable and the dependent variable. In our specific example, the slope coefficient for HSG is 0.6615.
This means that with every one-unit increase in High School Grade, the CPI increases by approximately 0.6615 units, holding all other variables constant.
If this coefficient were zero, it would suggest that changes in the High School Grade have no effect on the CPI.
Thus, the value of the slope coefficient is crucial as it directly influences the predictions made by the regression model.
By assessing its significance through hypothesis testing, we bring statistical rigor into understanding these relationships.

P-Value

The P-value is a key concept in hypothesis testing as it helps us understand the strength of the evidence against the null hypothesis. In this regression analysis, a P-value of 0.000 for HSG indicates it has a highly significant impact on CPI.
A low P-value (usually less than 0.05) means the null hypothesis is unlikely, and we have sufficient evidence to reject it in favor of the alternative hypothesis.
In simpler terms, the smaller the P-value, the stronger the evidence against the null. For our data, a P-value of 0.000 suggests that changes in High School Grade significantly affect the College Performance Index, with $<0.05$ suggesting a less than 5% chance that this relationship is due to random variation.

Linear Regression

Linear regression is a statistical technique used to model and analyze the relationships between dependent and independent variables. In the given scenario, the regression equation is:
\[ \text{CPI} = 1.1362 + 0.6615 \text{HSG} + 0.2301 \text{AMAP} + 0.0075 \text{ADST} \]
This equation predicts the College Performance Index (CPI) based on variables like High School Grade (HSG), Average Monthly Attendance Percentage (AMAP), and Average Daily Study Time (ADST).
The intercept (1.1362) represents the expected CPI when all independent variables are zero, although some values like AMAP and ADST may not realistically reach zero. Each slope coefficient shows how much the CPI changes with a one-unit change in that particular variable, assuming other variables stay constant.
Linear regression lets us quantify these relationships, helping in decision-making based on predictive analytics.