Introductory Statistics (2nd Edition)
2nd Edition
ISBN: 9780321978271
Author: Robert Gould, Colleen N. Ryan
Publisher: PEARSON
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Chapter 6, Problem 46SE
Males' SAT Scores According to the College Board, the
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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 6 Solutions
Introductory Statistics (2nd Edition)
Ch. 6 - 6.1-6.4 Directions Determine whether each of the...Ch. 6 - 6.1-6.4 Directions Determine whether each of the...Ch. 6 - 6.1-6.4 Directions Determine whether each of the...Ch. 6 - 6.1-6.4 Directions Determine whether each of the...Ch. 6 - Loaded Die (Example 2) A magician has shaved an...Ch. 6 - Prob. 6SECh. 6 - Distribution of Two Thumbtacks When a certain type...Ch. 6 - Distribution of Two Coin Flips When a fair coin is...Ch. 6 - Two Thumbtacks a. From your answers in Exercise...Ch. 6 - Two Coins a. From your answers in Exercise 6.8,...
Ch. 6 - Snow Depth (Example 3) Eric wants to go skiing...Ch. 6 - Snow Depth Refer to Exercise 6.11. What is the...Ch. 6 - Applying the Empirical Rule with z-Scores The...Ch. 6 - IQs Wechsler IQs are approximately Normally...Ch. 6 - SAT Scores Quantitative SAT scores are...Ch. 6 - Women’s Heights Assume that college women’s...Ch. 6 - Women’s height (Example 4) College women have a...Ch. 6 - Act scores ACT score are approximately Normally...Ch. 6 - Standard Normal Use the table or technology to...Ch. 6 - Standard Normal Use a table or technology to...Ch. 6 - Standard Normal Use a table or technology to...Ch. 6 - Standard Normal Use a table or technology to...Ch. 6 - Extreme Positive z -Scores For each question, find...Ch. 6 - Extreme Negative z-Scores For each question, find...Ch. 6 - Females' SAT Scores (Example 5) According to data...Ch. 6 - Males' SAT Scores According to data from the...Ch. 6 - Stanford-Binet IQs Stanford-Binet IQ scores for...Ch. 6 - Stanford-Binet IQs Stanford-Binet IQs for children...Ch. 6 - Birth Length (Example 6) According to National...Ch. 6 - White Blood Cells The distribution of white blood...Ch. 6 - Red Blood Cells: Men The distribution of red blood...Ch. 6 - Red Blood Cells: Women Answer the previous...Ch. 6 - SAT Scores in Alaska In Alaska in 2010, the...Ch. 6 - SAT Scores in Connecticut In Connecticut in 2010,...Ch. 6 - SAT Scores in New Jersey In New Jersey in 2010,...Ch. 6 - SAT Scores in Texas In Texas in 2010, the average...Ch. 6 - New York City Weather New York City’s mean minimum...Ch. 6 - Women's Heights Assume for this question that...Ch. 6 - Probability or Measurement (Inverse)? (Example 7)...Ch. 6 - Probability or Measurement (Inverse)? The Normal...Ch. 6 - Inverse Normal, Standard In a standard Normal...Ch. 6 - Inverse Normal, Standard In a standard Normal...Ch. 6 - Inverse Normal, Standard Assume a standard Normal...Ch. 6 - Inverse Normal, Standard Assume a standard Normal...Ch. 6 - Females' SAT Scores (Example 8) According to the...Ch. 6 - Males' SAT Scores According to the College Board,...Ch. 6 - Tall Club, Women Suppose there is a club for tall...Ch. 6 - Tall Club, Men Suppose there is a club for tall...Ch. 6 - Women’s Heights Suppose college women’s heights...Ch. 6 - Men’s Heights Suppose college men’s heights are...Ch. 6 - Inverse SATs Critical reading SAT scores are...Ch. 6 - Inverse Women’s Heights College women have heights...Ch. 6 - Girls’ and Women’s Heights According to the...Ch. 6 - Boys’ and Men’s Heights According to the National...Ch. 6 - Cats’ Birth Weights The average birth weight of...Ch. 6 - Elephants’ Birth Weights The average birth weight...Ch. 6 - Gender of Children (Example 10) A married couple...Ch. 6 - Coin Flip A coin will be flipped four times, and...Ch. 6 - Coin Flips (Example 10) A teacher wants to find...Ch. 6 - Twins In Exercise 6.59 you are told to assume that...Ch. 6 - Divorce Suppose that the probability that a...Ch. 6 - Divorce Suppose that the probability that a...Ch. 6 - Identifying n, p, and x (Example 11) For each...Ch. 6 - Identifying n, p, and x For each situation,...Ch. 6 - Stolen Bicycles (Example 12) According to the...Ch. 6 - Florida Recidivism Rate The three-year recidivism...Ch. 6 - Prob. 67SECh. 6 - Cornell Admission The undergraduate admission rate...Ch. 6 - Wisconsin Graduation Wisconsin has the highest...Ch. 6 - Colorado Graduation Colorado has a high school...Ch. 6 - Florida Homicide Clearance The homicide clearance...Ch. 6 - Virginia Homicide Clearance The homicide clearance...Ch. 6 - DWI Convictions (Example 13) In New Mexico, about...Ch. 6 - Internet Access A 2013 Gallup poll indicated that...Ch. 6 - Drunk Walking You may have heard that drunk...Ch. 6 - Texting While Driving According to a Pew poll in...Ch. 6 - Coin Flip (Example 14) A fair coin is flipped 50...Ch. 6 - Drivers Aged 60-65 According to GMAC Insurance, 20...Ch. 6 - Prob. 79SECh. 6 - Prob. 80SECh. 6 - Birth Length A study of U.S. births published on...Ch. 6 - Birth Length A study of U.S. births published on...Ch. 6 - Males’ Body Temperatures A study of human body...Ch. 6 - Females’ Body Temperatures A study of human body...Ch. 6 - Prob. 85CRECh. 6 - Cremation Rates in Mississippi, Binomial and...Ch. 6 - Prob. 87CRECh. 6 - Quantitative SAT Scores, Normal and Binomial The...Ch. 6 - Prob. 89CRECh. 6 - Birth Length and z-Scores, Inverse Babies in the...
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