A Statistics Professor Gave Three Quizzes Leading Up to the First

question 72

Essay

A statistics professor gave three quizzes leading up to the first test in his class. The quiz grades and test grade for each of eight students are given in the table.

$\begin{array}{ccccc}\hline \text { Student Test Grade } & \text { Quiz } 1 & \text { Quiz } 2 & \text { Quiz } 3 \\\hline 1 & 75 & 8 & 9 & 5 \\2 & 89 & 10 & 7 & 6 \\3 & 73 & 9 & 8 & 7 \\4 & 91 & 8 & 7 & 10 \\5 & 64 & 9 & 6 & 6 \\6 & 78 & 8 & 7 & 6 \\7 & 83 & 10 & 8 & 7 \\8 & 71 & 9 & 4 & 6 \\\hline\end{array}$
The professor fit a first-order model to the data that he intends to use to predict a student's grade on the first test using that student's grades on the first three quizzes.

Test the null hypothesis $H _ { 0 } : \beta _ { 1 } = \beta _ { 2 } = \beta _ { 3 } = 0$ against the alternative hypothesis $H _ { \mathrm { a } }$ : at least one $\beta _ { i } \neq 0$ . Use $\alpha = .05$ . Interpret the result.

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Definitions:

Northern State

A reference to one of the states in the northern part of the United States, often in the context of its opposition to slavery during the Civil War era.

Indentured Servants

Individuals who agreed to work for a specific number of years in exchange for passage to the New World and eventual freedom.

Gradual Emancipation

A legal process implemented in various jurisdictions to end slavery over time, often by stipulating that children born to enslaved mothers would be free after a certain age.

Mid-Atlantic Region

An area of the United States that includes states like New York, New Jersey, and Pennsylvania, known for its significant cultural and economic contribution.