Question

At a large university, the Statistics Department has tried a different text during each of the last three quarters. During the fall quarter, 500 students used a book by Professor Mean; during the winter quarter, 300 students used a book by Professor Median; and during the spring quarter, 200 students used a book by Professor Mode. A survey at the end of each quarter showed that 200 students were satisfied with the text in the fall quarter, 150 in the winter quarter, and 160 in the spring quarter. a. If a student who took statistics during one of these three quarters is selected at random, what is the probability that the student was satisfied with the textbook? b. If a randomly selected student reports being satisfied with the book, is the student most likely to have used the book by Mean, Median, or Mode? Who is the least likely author? (Hint: Use Bayes' rule to compute three probabilities.)

Short answer

The probability that a student is satisfied with the textbook is 51%. If a student reports being satisfied with the book, they are most likely to have used the book by Mean (39.2% probability), followed by Mode (31.4% probability). The least likely author is Median (29.4% probability).

Step-by-step solution

Calculate Total Students and Satisfaction Rates

First, calculate the total number of students who took part in the study across the three quarters. This is done by adding up all the students: 500 (Fall) + 300 (Winter) + 200 (Spring) = 1000 students. Next, calculate the total number of satisfied students during the same period: 200 (Fall) + 150 (Winter) + 160 (Spring) = 510 satisfied students. With this, calculate the overall satisfaction rate by dividing the total number of satisfied students by the total number of students: \(\frac{510}{1000} = 0.51\), or 51%.

Calculate Quarter-wise Satisfaction Rates

Now, calculate the satisfaction rates for each individual quarter. For Fall, it would be \(\frac{200}{500}=0.4\) or 40%. For Winter, \(\frac{150}{300}=0.5\) or 50%. And, for Spring, \(\frac{160}{200}=0.8\) or 80%.

Calculate Probabilities Using Bayes' Rule

Next, use Bayes' rule to calculate the probability that a satisfied student was from a particular quarter. Bayes' rule is expressed as \(P(A|B) = \frac{P(B|A)*P(A)}{P(B)}\), where A and B are events, P(A) is the independent probability of A, P(B) is the independent probability of B, P(A|B) is the posterior probability of A given B, and P(B|A) is the likelihood of B given A. In this case, A is 'student was in a particular quarter' and B is 'student was satisfied'. For a student from the Fall quarter, for example, this would be calculated as follows: \(P(Fall|Satisfied) = \frac{(0.4)*(0.5)}{(0.51)} = 0.392157\), or 39.2%. Similarly, for Winter: \(P(Winter|Satisfied) = \frac{(0.5)*(0.3)}{(0.51)} = 0.294118\), or 29.4%. And for Spring: \(P(Spring|Satisfied) = \frac{(0.8)*(0.2)}{(0.51)} = 0.313725\), or 31.4%.

Key concepts

Probability Calculation

Understanding the principles of probability calculation is essential for students diving into the realm of statistics, where quantifying likelihood is the backbone of data analysis. Let's explore an everyday scenario: students evaluating their satisfaction with textbooks over different academic quarters. To comprehend the overall contentment, we need to calculate the probability of a student being satisfied with the textbook.

The process begins by identifying the total population—in our example, all the students who participated across three quarters—amounting to 1,000 students. Subsequently, we delve into the count of satisfied students, which is 510. To grasp the probability of satisfaction, divide the number of satisfied students by the total number. Mathematically, we express it as \(\frac{510}{1000} = 0.51\), indicating a 51% chance of student satisfaction.

This calculation embodies the probability's foundation and sets the stage for more sophisticated analyses, such as deducing probabilities for specific circumstances using tools like Bayes' rule.

Bayes' Rule Application

Mastering Bayes' rule is a statistical game-changer, especially when dissecting conditional probabilities—figuring out the chance of an event given another has already occurred. It's particularly useful in our textbook satisfaction analysis.

Bayes' theorem can be articulated as \(P(A|B) = \frac{P(B|A)\times P(A)}{P(B)}\), where \P(A)\ and \P(B)\ are the independent probabilities of events A and B respectively, \P(A|B)\ is the probability of A given B, and \P(B|A)\ is the probability of B given A.

In our situation, we're keen to see if a satisfied student likely used a book by Professor Mean, Median, or Mode. By applying Bayes' rule and plugging in our quarters' satisfaction rates, we can deduce this precisely. For instance, for a student from the Fall quarter, the calculation yields \(P(Fall|Satisfied) = \frac{(0.4)\times(0.5)}{(0.51)} = 0.392157\), or approximately 39.2%. With similar computations, we paint a clear picture of which professor's textbook likely resulted in student satisfaction—crucial insights for educational strategists.

Satisfaction Rates Analysis

Scrutinizing satisfaction rates is a pivotal part of educational feedback systems, providing insights that assist in making informed decisions about academic resources. Our statistics textbook case study showcases this analysis by examining the satisfaction of students with three distinct textbooks across consecutive quarters.

The satisfaction rates are isolated for each quarter: 40% for Professor Mean (Fall), 50% for Professor Median (Winter), and an impressive 80% for Professor Mode (Spring). These figures are pivotal—they not only reflect the students' approval but when juxtaposed, they reveal a compelling story about teaching quality and resource effectiveness over time. We recognize that Professor Mode's textbook, used during the spring quarter, has the highest satisfaction rate. This insight suggests the importance of the right educational materials in student success and contentment.

Analyzing satisfaction rates isn't solely about percentages—it's a deep dive into the effectiveness of academic strategies and their impacts on the learning journey.