91Ó°ÊÓ

Are people more confident in their answers when the answer is actually correct than when it is not? The article "Female Students Less Confident, More Accurate Than Male Counterparts" described a study that measured medical students" confidence and the accuracy of their responses. Participants categorized their confidence levels using either "sure," "feeling lucky," or "no clue" Define the following events: \(C=\) event that a response is conect \(S=\) event that confidence level is "sure" \(L=\) event that confidence level is "feeling lucky" \(N=\) event that confidence level is "no clue" a. Data from the article were used to estimate the following probabilities for males: $$ \begin{array}{rrr} P(S)=0.442 & P(L)=0.422 & P(N)=0.136 \\ P(C \mid S)=0.783 & P(C \mid L)=0.498 & P(C \mid N)=0.320 \end{array} $$ Use the given probabilities to construct a hypothetical 1000 table with rows corresponding to confidence level and columns comesponding to whether the response was correct or not. b. Calculate the probability that a male student's confidence level is "sure" given that the response is correct. c. Calculate the probability that a male student's confidence level is "no clue" given that the response is incorrect. d. Calculate the probability that a male student's response is correct. e. Data from the article were also used to estimate the following probabilities for females: $$ \begin{array}{rrr} P(S)=0.395 & P(L)=0.444 & P(N)=0.161 \\ P(C 1 S)=0.805 & P(C \mid L)=0.535 & P(C \mid N)=0.320 \end{array} $$ Use the given probabilities to construct a hypothetical 1000 table with rows corresponding to confidence level and columns corresponding to whether the response was comect or not. f. Calculate the probability that a female student's confidence level is "sure" given that the response is correct. g. Calculate the probability that a female student's confidence level is "no clue" given that the response is incorrect. h. Calculate the probability that a female student's response is correct. i. Do the given probabilities and the probabilities that you calculated support the statement in the title of the article? Explain.

Step by step solution

01

Calculate Correct and Incorrect Probabilities

We will first calculate the probabilities of each event (correct and incorrect) given the confidence levels S, L, and N. For males: Using the given data, $$ \begin{aligned} P(C | S) &= 0.783 \\ P(C | L) &= 0.498 \\ P(C | N) &= 0.320 \end{aligned} $$ We can calculate the probability of incorrect responses given the confidence levels (I) using the fact that the total probability is 1: $$ \begin{aligned} P(I | S) &= 1 - P(C | S) = 0.217 \\ P(I | L) &= 1 - P(C | L) = 0.502 \\ P(I | N) &= 1 - P(C | N) = 0.680 \end{aligned} $$

02

Construct Hypothetical Table

We will now use these probabilities to construct the table for 1000 male students. $$ \begin{array}{c|c|c|c} Confidence Level & Correct & Incorrect & Total \\ \hline Sure & 0.442 \times 0.783 \times 1000 & 0.442 \times 0.217 \times 1000 & 0.442 \times 1000 \\ \hline Feeling\ Lucky & 0.422 \times 0.498 \times 1000 & 0.422 \times 0.502 \times 1000 & 0.422 \times 1000 \\ \hline No\ Clue & 0.136 \times 0.320 \times 1000 & 0.136 \times 0.680 \times 1000 & 0.136 \times 1000 \\ \end{array} $$ To find the numbers in the table, multiply the probability of each event by 1000. b. Calculate the probability that a male student's confidence level is "sure" given that the response is correct.

03

Apply Bayes' theorem

We will use Bayes' theorem to find the probability that a male student's confidence level is "sure" given that the response is correct: $$ P(S | C) = \frac{P(C | S) \times P(S)}{P(C)} $$ We already have the probabilities \(P(C | S) = 0.783\) and \(P(S) = 0.442\). We need to calculate \(P(C)\).

04

Calculate P(C)

We will use the law of total probability to find \(P(C)\): $$ P(C) = P(C | S) \times P(S) + P(C | L) \times P(L) + P(C | N) \times P(N) $$ Plugging in the probabilities we have: $$ P(C) = (0.783 \times 0.442) + (0.498 \times 0.422) + (0.320 \times 0.136) $$ Now, we can calculate \(P(S | C)\): $$ P(S | C) = \frac{0.783 \times 0.442}{P(C)} $$ c. Calculate the probability that a male student's confidence level is "no clue" given that the response is incorrect. Similarly, to find the probability that a male student's confidence level is "no clue" given that the response is incorrect, we apply Bayes' theorem: $$ P(N | I) = \frac{P(I | N) \times P(N)}{P(I)} $$ We already know \(P(I | N) = 0.680\) and \(P(N) = 0.136\). To calculate \(P(I)\), we can use the fact that \(P(I) = 1 - P(C)\). Now, we can calculate \(P(N | I)\): $$ P(N | I) = \frac{0.680 \times 0.136}{P(I)} $$ I will leave the calculations for the female students (steps d to h) for you to practice. You should follow the same method applied for male students. i. Do the given probabilities and the probabilities that you calculated support the statement in the title of the article? After calculating all the probabilities for both male and female students, we will compare their confidence levels and accuracy to analyze the statement in the title of the article. Look at the probabilities calculated for males and females and compare them to see if the data supports the statement "Female Students Less Confident, More Accurate Than Male Counterparts." It would help to note the differences in the conditional probabilities calculated for each gender and how they relate to confidence levels and accuracy.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Understanding Bayes' Theorem

Bayes' Theorem is vital for understanding how the likelihood of an event changes with the acquisition of new information. Take, for instance, a medical student answering a question, where their confidence level can be 'sure', 'feeling lucky', or 'no clue'. Bayes' Theorem allows us to update the probability that their answer is correct, based on their asserted confidence level.

Mathematically, Bayes' theorem is expressed as:
\[P(A | B) = \frac{P(B | A) \times P(A)}{P(B)}\]
Where:

• \( P(A | B) \) is the posterior probability: the chance of event A occurring given that B is true.
• \( P(B | A) \) is the likelihood: the chance of observing B given event A.
• \( P(A) \) is the prior probability: the initial degree of belief in event A.
• \( P(B) \) is the marginal probability: the total probability that event B will occur regardless of A.

In the context of the exercise, we calculate the probability that a male student is 'sure' about their answer given that the response is correct using Bayes' Theorem. It combines what we know about general confidence levels among students with the likelihoods of correctness based on confidence.

The Total Probability Theorem

When we consider the total probability of an event, we account for all possible ways that event can occur. The Law of Total Probability is essential when an event can be partitioned into a set of mutually exclusive events, which the exercise illustrates with confidence levels.

The theorem is written as:
\[P(A) = \sum_{i=1}^{n} P(A | B_i)P(B_i)\]
Here, \(P(A)\) is the probability of event A, \(P(A | B_i)\) is the probability of event A given \(B_i\), and \(P(B_i)\) is the probability of event \(B_i\). We sum over all possible \(B_i\) events.

For instance, the overall probability of a correct answer, \(P(C)\), is found by considering the likelihood of being correct with each confidence level and the overall occurrence rates of these confidence levels. The probabilities of being correct when ‘sure’, ‘feeling lucky’, or having ‘no clue’ are each weighed by the prevalence of these confidence states among students.

Confidence Level in Statistics

The term 'confidence level' in statistics generally pertains to the confidence intervals used for parameter estimation. However, in the context of our exercise, it describes the self-assessed certainty a student has regarding their answer's correctness. This subjective measure is critical because it serves as an indicator of performance and can be correlated with the actual correctness of their answers.

Statistically, by calculating the conditional probabilities for each confidence level, given the correctness or incorrectness of a response, we can gauge the reliability of a student's self-assessment. Higher confidence levels are frequently assumed to be correlated with higher accuracy, but this must be verified by analysis. In the textbook problem, using the total probability theorem and Bayes' theorem, we can compare these confidence levels with the actual probabilities of a correct answer, thus assessing the validity of the initial statement regarding students' confidence and accuracy.