91Ó°ÊÓ

Five hundred first-year students at a state university were classified according to both high school GPA and whether they were on academic probation at the end of their first semester. The data are $$ \begin{array}{lcccc} && {\text { High School GPA }} \\ & 2.5 \text { to } & 3.0 \text { to } & 3.5 \text { and } & \\ \text { Probation } & <3.0 & <3.5 & \text { Above } & \text { Total } \\ \hline \text { Yes } & 50 & 55 & 30 & 135 \\ \text { No } & 45 & 135 & 185 & 365 \\ \text { Total } & 95 & 190 & 215 & 500 \\ \hline \end{array} $$ a. Construct a table of the estimated probabilities for each GPA-probation combination. b. Use the table constructed in Part (a) to approximate the probability that a randomly selected first-year student at this university will be on academic probation at the end of the first semester. c. What is the estimated probability that a randomly selected first-year student at this university had a high school GPA of \(3.5\) or above? d. Are the two outcomes selected student has a bigh school GPA of \(3.5\) or above and selected student is on academic probation at the end of the first semester independent outcomes? How can you tell? e. Estimate the proportion of first-year students with high school GPAs between \(2.5\) and \(3.0\) who are on academic probation at the end of the first semester. f. Estimate the proportion of those first-year students with high school GPAs \(3.5\) and above who are on academic probation at the end of the first semester.

Step by step solution

01

Construct the table of estimated probabilities

Each cell within the table represents the joint probability of the two outcomes \(P(A \cap B) = \frac{n(A \cap B)}{n(S)}\) where \(n(A \cap B)\) represents the intersection count and \(n(S)\) represents the total count. For instance, the probability of a student with a GPA of less than 3.0 being on probation is \(\frac{50}{500} = 0.1\). All the other probabilities can be calculated in a similar way.

02

Probablity for a student to be on probation

The probability of a student being on probation \(P(Probation) = \frac{n(Probation)}{n(S)}\). From the table, we know that the total number of students on probation is 135. So, the probability will be \(\frac{135}{500} = 0.27\).

03

Probablity for a student to have a high school GPA of 3.5 or above

Probability \(P(GPA \geq 3.5) = \frac{n(GPA \geq 3.5)}{n(S)}\). From the table, we can see that the total number of students with a GPA of 3.5 and above is 215, hence \(\frac{215}{500} = 0.43\).

04

Check for independence of two events

Two events A & B are said to be independent if \(P(A \cap B) = P(A)P(B)\). We compute \(P(GPA \geq 3.5 \cap Probation) = P(GPA \geq 3.5)P(Probation)\). If these values are equal, the events are independent; otherwise, they are dependent.

05

Estimate the proportion of first-year students with high school GPAs between 2.5 and 3.0

For students with GPAs between 2.5 and 3.0, we can calculate the proportion of students on probation as \(P(Probation | GPA = 2.5-3.0) = \frac{n(Probation \cap GPA = 2.5-3.0)}{n(GPA = 2.5-3.0)} = \frac{50}{95} = 0.53\).

06

Estimate the proportion of those first-year students with high school GPAs 3.5 and above

For students with GPAs 3.5 and above, we can calculate the proportion of students on probation as \(P(Probation | GPA \geq 3.5) = \frac{n(Probation \cap GPA \geq 3.5)}{n(GPA \geq 3.5)} = \frac{30}{215} = 0.14 \).

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Key Concepts

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

Joint Probability

Joint Probability refers to the probability of two events occurring simultaneously. It helps in understanding how likely two scenarios are to happen at the same time. In our exercise, we look at students' high school GPA and whether they are on academic probation simultaneously. Here's how we assess joint probability:

• We determine it by dividing the frequency of both events happening by the total number of observations.
• This is the probability of a specific GPA bracket and being on probation occurring together.

For example, there are 50 students with a GPA below 3.0 on probation. The joint probability is \[P\left(GPA < 3.0 \cap \text{Probation}\right) = \frac{50}{500} = 0.1 \]Hence, there's a 10% chance a random student will both have a GPA less than 3.0 and be on probation.

Conditional Probability

Conditional Probability is the likelihood of an event occurring given that another event has occurred. It provides insights into how one condition affects the occurrence of another condition, which is essential in checking dependencies between events. It's especially useful in analyzing patterns within datasets.For instance, if we want to know the probability of a student being on probation given they have a GPA between 2.5 and 3.0, we use the formula:\[P(\text{Probation} | GPA = 2.5-3.0) = \frac{P(GPA = 2.5-3.0 \cap \text{Probation})}{P(GPA = 2.5-3.0)}\]Calculation:- Number on probation (GPA 2.5-3.0) = 50- Total in GPA range 2.5-3.0 = 95\[P(\text{Probation} | GPA = 2.5-3.0) = \frac{50}{95} \approx 0.53\]Therefore, there's approximately a 53% chance a student with a GPA between 2.5 and 3.0 will be on probation.

Independence of Events

The concept of independence between two events in probability suggests that the occurrence of one event does not affect the occurrence of the other. For two events A and B to be independent, the following must hold true:\[P(A \cap B) = P(A)P(B)\]In our example, we are examining whether having a GPA of 3.5 and above is independent of being on academic probation.Check for Independence:

• Calculate the individual probability of GPA ≥ 3.5:
\[P(GPA \geq 3.5) = \frac{215}{500} = 0.43\]
• Calculate the individual probability of being on probation:
\[P(\text{Probation}) = \frac{135}{500} = 0.27\]
• Calculate the joint probability of GPA ≥ 3.5 and on probation:
\[P(GPA \geq 3.5 \cap \text{Probation}) = \frac{30}{500} = 0.06\]Since 0.43 * 0.27 ≠ 0.06, the events are not independent. Therefore, GPA ≥ 3.5 does influence the likelihood of probation.

High School GPA Analysis

Analyzing students' high school GPA can provide valuable insights into their academic journeys and likelihood of facing probation. By breaking down GPAs into categories, institutions can offer targeted support to students who may be at risk. Analysis Steps:

• We categorize students based on their high school GPA ranges: below 3.0, between 3.0-3.5, and above 3.5.
• We then assess how many in each category are on academic probation.

Findings: For students with a GPA:

• < 3.0, roughly 52.6% (50 out of 95) are on probation.
• Between 3.0 and 3.5, roughly 28.9% (55 out of 190) are on probation.
• 3.5 and above, only about 14% (30 out of 215) are on probation.

This pattern suggests a trend where students with lower high school GPAs are more susceptible to academic probation, highlighting the importance of early support interventions.