Students in a business statistics class were asked what...

Students in a business statistics class were asked what grade they expected in the course and whether they worked on additional problems beyond those assigned by the instructor. The following table gives proportions of students in each of eight joint classifications. $$ \begin{array}{lcccc} \hline \text { Worked } & & {\text { Expected Grade }} \\ \text { Problems } & \mathrm{A} & \mathrm{B} & \mathrm{C} & \text { Below C } \\ \hline \text { Yes } & 0.12 & 0.06 & 0.12 & 0.02 \\ \text { No } & 0.13 & 0.21 & 0.26 & 0.08 \\ \hline \end{array} $$ a. Find the probability that a randomly chosen student from this class worked on additional problems. b. Find the probability that a randomly chosen student from this class expects an $A$. c. Find the probability that a randomly chosen student who worked on additional problems expects an A. d. Find the probability that a randomly chosen student who expects an A worked on additional problems. e. Find the probability that a randomly chosen student who worked on additional problems expects a grade below $\mathrm{B}$. f. Are "worked additional problems" and "expected grade" statistically independent?

Students in a business statistics class were asked what grade they expected in the course and whether they worked on additional problems beyond those assigned by the instructor. The following table gives proportions of students in each of eight joint classifications.
$$
\begin{array}{lcccc}
\hline \text { Worked } & & {\text { Expected Grade }} \\
\text { Problems } & \mathrm{A} & \mathrm{B} & \mathrm{C} & \text { Below C } \\
\hline \text { Yes } & 0.12 & 0.06 & 0.12 & 0.02 \\
\text { No } & 0.13 & 0.21 & 0.26 & 0.08 \\
\hline
\end{array}
$$
a. Find the probability that a randomly chosen student from this class worked on additional problems.
b. Find the probability that a randomly chosen student from this class expects an $A$.
c. Find the probability that a randomly chosen student who worked on additional problems expects an A.
d. Find the probability that a randomly chosen student who expects an A worked on additional problems.
e. Find the probability that a randomly chosen student who worked on additional problems expects a grade below $\mathrm{B}$.
f. Are "worked additional problems" and "expected grade" statistically independent?

Key Concepts

Statistical Independence

Statistical independence between two events or variables means that the occurrence of one does not affect the probability of the occurrence of the other. In the context of contingency tables, two variables are independent if and only if the joint probability of any pair of outcomes equals the product of their marginal probabilities.

Marginal Probability

Marginal probability is the probability of an event occurring irrespective of the outcomes of other variables. In a contingency table, it is calculated by summing the joint probabilities across the rows or columns, thus giving the overall probability for one of the variables.

Conditional Probability

Conditional probability is the likelihood of an event occurring given that another event has already occurred. This concept is used to update the probability of an event based on additional information, and in a contingency table, it is computed by taking the ratio of the joint probability of both events to the marginal probability of the given condition.

Contingency Table

A contingency table is a tabular representation of data that shows the frequency distribution of variables. It is used to display the joint distribution of two or more categorical variables, allowing analysts to observe the relationships and interactions between them.

Joint Probability

Joint probability refers to the probability of two or more events happening simultaneously. In the context of a contingency table, it corresponds to the proportion or count in a specific cell that represents the simultaneous occurrence of particular values for each variable.