Essential Statistics
2nd Edition
ISBN: 9781259570643
Author: Navidi
Publisher: MCG
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Chapter 5.1, Problem 37E
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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.
5. Probability Distributions – Continuous Random Variables
A factory machine produces metal rods whose lengths (in cm) follow a continuous uniform distribution on the interval [98, 102].
Questions:
a) Define the probability density function (PDF) of the rod length.b) Calculate the probability that a randomly selected rod is shorter than 99 cm.c) Determine the expected value and variance of rod lengths.d) If a sample of 25 rods is selected, what is the probability that their average length is between 99.5 cm and 100.5 cm? Justify your answer using the appropriate distribution.
Chapter 5 Solutions
Essential Statistics
Ch. 5.1 - 1. A family has three children. If the genders of...Ch. 5.1 - 2. Someone says that the following table shows the...Ch. 5.1 - Prob. 3CYUCh. 5.1 - 4. Following is the probability distribution of a...Ch. 5.1 - Prob. 5CYUCh. 5.1 - Prob. 6CYUCh. 5.1 - Prob. 7CYUCh. 5.1 - 8. Following is the probability distribution for...Ch. 5.1 - Prob. 9ECh. 5.1 - In Exercises 9–12, fill in each blank with the...
Ch. 5.1 - Prob. 11ECh. 5.1 - Prob. 12ECh. 5.1 - Prob. 13ECh. 5.1 - Prob. 14ECh. 5.1 - Prob. 15ECh. 5.1 - Prob. 16ECh. 5.1 - Prob. 17ECh. 5.1 - In Exercises 17–26, determine whether the random...Ch. 5.1 - Prob. 19ECh. 5.1 - In Exercises 17–26, determine whether the random...Ch. 5.1 - Prob. 21ECh. 5.1 - In Exercises 17–26, determine whether the random...Ch. 5.1 - Prob. 23ECh. 5.1 - Prob. 24ECh. 5.1 - Prob. 25ECh. 5.1 - Prob. 26ECh. 5.1 - In Exercises 27–32, determine whether the table...Ch. 5.1 - In Exercises 27–32, determine whether the table...Ch. 5.1 - Prob. 29ECh. 5.1 - In Exercises 27–32, determine whether the table...Ch. 5.1 - Prob. 31ECh. 5.1 - In Exercises 27–32, determine whether the table...Ch. 5.1 - Prob. 33ECh. 5.1 - In Exercises 33–38, compute the mean and standard...Ch. 5.1 - Prob. 35ECh. 5.1 - In Exercises 33–38, compute the mean and standard...Ch. 5.1 - In Exercises 33–38, compute the mean and standard...Ch. 5.1 - In Exercises 33–38, compute the mean and standard...Ch. 5.1 - Prob. 39ECh. 5.1 - 40. Fill in the missing value so that the...Ch. 5.1 - 41. Put some air in your tires: Let X represent...Ch. 5.1 - Prob. 42ECh. 5.1 - Prob. 43ECh. 5.1 - Prob. 44ECh. 5.1 - Prob. 45ECh. 5.1 - Prob. 46ECh. 5.1 - Prob. 47ECh. 5.1 - 48. Pain: The General Social Survey asked 827...Ch. 5.1 - Prob. 49ECh. 5.1 - Prob. 50ECh. 5.1 - 51. Lottery: In the New York State Numbers...Ch. 5.1 - 52. Lottery: In the New York State Numbers...Ch. 5.1 - Prob. 53ECh. 5.1 - Prob. 54ECh. 5.1 - Prob. 55ECh. 5.1 - Prob. 56ECh. 5.1 - Prob. 57ECh. 5.1 - Prob. 58ECh. 5.1 - Prob. 59ECh. 5.1 - Prob. 60ECh. 5.1 - Prob. 61ECh. 5.2 - 1. Determine whether X is a binomial random...Ch. 5.2 - Prob. 2CYUCh. 5.2 - Prob. 3CYUCh. 5.2 - Prob. 4CYUCh. 5.2 - Prob. 5ECh. 5.2 - In Exercises 5–7, fill in each blank with the...Ch. 5.2 - Prob. 7ECh. 5.2 - In Exercises 8–10, determine whether the statement...Ch. 5.2 - Prob. 9ECh. 5.2 - In Exercises 8–10, determine whether the statement...Ch. 5.2 - Prob. 11ECh. 5.2 - Prob. 12ECh. 5.2 - Prob. 13ECh. 5.2 - In Exercises 11–16, determine whether the random...Ch. 5.2 - Prob. 15ECh. 5.2 - In Exercises 11–16, determine whether the random...Ch. 5.2 - Prob. 17ECh. 5.2 - In Exercises 17–26, determine the indicated...Ch. 5.2 - Prob. 19ECh. 5.2 - In Exercises 17–26, determine the indicated...Ch. 5.2 - In Exercises 17–26, determine the indicated...Ch. 5.2 - Prob. 22ECh. 5.2 - Prob. 23ECh. 5.2 - In Exercises 17–26, determine the indicated...Ch. 5.2 - Prob. 25ECh. 5.2 - Prob. 26ECh. 5.2 - Prob. 27ECh. 5.2 - 28. Take another guess: A student takes a...Ch. 5.2 - Prob. 29ECh. 5.2 - Prob. 30ECh. 5.2 - Prob. 31ECh. 5.2 - 32. What should I buy? A study conducted by the...Ch. 5.2 - Prob. 33ECh. 5.2 - Prob. 34ECh. 5.2 - Prob. 35ECh. 5.2 - Prob. 36ECh. 5.2 - Prob. 37ECh. 5.2 - 38. Stress at work: In a poll conducted by the...Ch. 5.2 - Prob. 39ECh. 5.2 - Prob. 40ECh. 5.2 - Prob. 41ECh. 5 - Prob. 1CQCh. 5 - Prob. 2CQCh. 5 - Prob. 3CQCh. 5 - Prob. 4CQCh. 5 - Prob. 5CQCh. 5 - Prob. 6CQCh. 5 - Prob. 7CQCh. 5 - Prob. 8CQCh. 5 - 9. At a cell phone battery plant, 5% of cell phone...Ch. 5 - Prob. 10CQCh. 5 - Prob. 11CQCh. 5 - Prob. 12CQCh. 5 - Prob. 13CQCh. 5 - Prob. 14CQCh. 5 - Prob. 15CQCh. 5 - Prob. 1RECh. 5 - Prob. 2RECh. 5 - Prob. 3RECh. 5 - Prob. 4RECh. 5 - Prob. 5RECh. 5 - 6. Lottery tickets: Refer to Exercise 5. What is...Ch. 5 - Prob. 7RECh. 5 - Prob. 8RECh. 5 - Prob. 9RECh. 5 - Prob. 10RECh. 5 - Prob. 11RECh. 5 - Prob. 12RECh. 5 - Prob. 13RECh. 5 - Prob. 14RECh. 5 - Prob. 15RECh. 5 - Prob. 1WAICh. 5 - Prob. 2WAICh. 5 - Prob. 3WAICh. 5 - Prob. 4WAICh. 5 - Prob. 5WAICh. 5 - Prob. 6WAICh. 5 - One of the most surprising probability...
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