Cervical Cancer (Example 18) According to a study published in Scientific American , about 8 women in 100,000 have cervical cancer (which we’ll call event C), so P C = 0.00008 . Suppose the chance that a Pap smear will detect cervical cancer when it is present is 0.84. Therefore, P(test pos | C) = 0.84 What is the probability that a randomly chosen woman who has this test will both have cervical cancer AND test positive for it?
Cervical Cancer (Example 18) According to a study published in Scientific American , about 8 women in 100,000 have cervical cancer (which we’ll call event C), so P C = 0.00008 . Suppose the chance that a Pap smear will detect cervical cancer when it is present is 0.84. Therefore, P(test pos | C) = 0.84 What is the probability that a randomly chosen woman who has this test will both have cervical cancer AND test positive for it?
Solution Summary: The author explains that the probability that a randomly chosen woman will have both- cervical cancer and positive test for it is 0.000067.
Cervical Cancer (Example 18) According to a study published in Scientific American, about 8 women in 100,000 have cervical cancer (which we’ll call event C), so
P
C
=
0.00008
. Suppose the chance that a Pap smear will detect cervical cancer when it is present is 0.84. Therefore,
P(test pos
|
C)
=
0.84
What is the probability that a randomly chosen woman who has this test will both have cervical cancer AND test positive for it?
3. Bayesian Inference – Updating Beliefs
A medical test for a rare disease has the following characteristics:
Sensitivity (true positive rate): 99%
Specificity (true negative rate): 98%
The disease occurs in 0.5% of the population.
A patient receives a positive test result.
Questions:
a) Define the relevant events and use Bayes’ Theorem to compute the probability that the patient actually has the disease.b) Explain why the result might seem counterintuitive, despite the high sensitivity and specificity.c) Discuss how prior probabilities influence posterior beliefs in Bayesian inference.d) Suppose a second, independent test with the same accuracy is conducted and is also positive. Update the probability that the patient has the disease.
4. Linear Regression - Model Assumptions and Interpretation
A real estate analyst is studying how house prices (Y) are related to house size in square feet (X). A simple
linear regression model is proposed:
The analyst fits the model and obtains:
•
Ŷ50,000+150X
YBoB₁X + €
•
R² = 0.76
• Residuals show a fan-shaped pattern when plotted against fitted values.
Questions:
a) Interpret the slope coefficient in context.
b) Explain what the R² value tells us about the model's performance.
c) Based on the residual pattern, what regression assumption is likely violated? What might be the
consequence?
d) Suggest at least two remedies to improve the model, based on the residual analysis.
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, statistics and related others by exploring similar questions and additional content below.
Discrete Distributions: Binomial, Poisson and Hypergeometric | Statistics for Data Science; Author: Dr. Bharatendra Rai;https://www.youtube.com/watch?v=lHhyy4JMigg;License: Standard Youtube License