Introduction To Probability And Statistics
15th Edition
ISBN: 9781337554428
Author: Mendenhall, William.
Publisher: Cengage Learning,
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Chapter 1, Problem 1CS
To determine
The variable measured in the survey and its properties.
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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.
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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 1 Solutions
Introduction To Probability And Statistics
Ch. 1.1 - Experimental UnitsDefine the experimental units...Ch. 1.1 - Experimental UnitsDefine the experimental units...Ch. 1.1 - Experimental UnitsDefine the experimental units...Ch. 1.1 - Experimental UnitsDefine the experimental units...Ch. 1.1 - Experimental UnitsDefine the experimental units...Ch. 1.1 - Qualitative or Quantitative? Are the variables in...Ch. 1.1 - Qualitative or Quantitative? Are the variables in...Ch. 1.1 - Qualitative or Quantitative? Are the variables in...Ch. 1.1 - Qualitative or Quantitative? Are the variables in...Ch. 1.1 - Discrete or Continuous? Are the variables in...
Ch. 1.1 - Discrete or Continuous? Are the variables in...Ch. 1.1 - Discrete or Continuous? Are the variables in...Ch. 1.1 - Discrete or Continuous? Are the variables in...Ch. 1.1 - Discrete or Continuous? Are the variables in...Ch. 1.1 - Discrete or Continuous? Are the variables in...Ch. 1.1 - Discrete or Continuous? Are the variables in...Ch. 1.1 - Discrete or Continuous? Are the variables in...Ch. 1.1 - Discrete or Continuous? Are the variables in...Ch. 1.1 - Populations or Samples? In Exercises 1922,...Ch. 1.1 - Populations or Samples? In Exercises 1922,...Ch. 1.1 - Populations or Samples? In Exercises 1922,...Ch. 1.1 - Populations or Samples? In Exercises 1922,...Ch. 1.1 - Parking on Campus Six vehicles selected from a...Ch. 1.1 - Past U.S. Presidents A data set gives the ages at...Ch. 1.1 - Voter Attitudes You are a candidate for your state...Ch. 1.1 - Cancer Survival Times A researcher wants to...Ch. 1.1 - New Teaching Methods A researcher wants to know...Ch. 1.2 - Pie and Bar Charts The data in Exercises 13...Ch. 1.2 - Pie and Bar Charts The data in Exercises 13...Ch. 1.2 - Pie and Bar Charts The data in Exercises 13...Ch. 1.2 - Groups of People Fifty people are grouped into...Ch. 1.2 - Presidential Popularity After the elections of...Ch. 1.2 - Presidential Popularity After the elections of...Ch. 1.2 - Presidential Popularity After the elections of...Ch. 1.2 - Presidential Popularity After the elections of...Ch. 1.2 - Presidential Popularity After the elections of...Ch. 1.2 - Prob. 10ECh. 1.2 - Prob. 11ECh. 1.2 - Prob. 12ECh. 1.2 - Prob. 13ECh. 1.2 - Prob. 14ECh. 1.2 - Prob. 15ECh. 1.2 - Back to Work How long does it take you to adjust...Ch. 1.2 - Prob. 17ECh. 1.2 - Prob. 18ECh. 1.2 - Prob. 19ECh. 1.2 - Prob. 20ECh. 1.2 - Prob. 21ECh. 1.3 - DotplotsConstruct a dotplot for the data given in...Ch. 1.3 - DotplotsConstruct a dotplot for the data given in...Ch. 1.3 - Stem and Leaf I Construct a stem and leaf plot for...Ch. 1.3 - Stem and Leaf I Construct a stem and leaf plot for...Ch. 1.3 - Stem and Leaf I Construct a stem and leaf plot for...Ch. 1.3 - Stem and Leaf II Use the following set of data...Ch. 1.3 - Stem and Leaf II Use the following set of data...Ch. 1.3 - Prob. 8ECh. 1.3 - Prob. 9ECh. 1.3 - Comparing Graphs A discrete variable can take on...Ch. 1.3 - Prob. 11ECh. 1.3 - Prob. 12ECh. 1.3 - Prob. 13ECh. 1.3 - Prob. 14ECh. 1.3 - Prob. 15ECh. 1.3 - Prob. 16ECh. 1.3 - Prob. 17ECh. 1.3 - Calcium Contents The calcium content (Ca)of a...Ch. 1.3 - Prob. 19ECh. 1.3 - Prob. 20ECh. 1.3 - Prob. 21ECh. 1.3 - Prob. 22ECh. 1.3 - Prob. 23ECh. 1.3 - Prob. 24ECh. 1.3 - American Presidents The following table lists the...Ch. 1.4 - Graphing Relative Frequency Histograms Construct a...Ch. 1.4 - Graphing Relative Frequency Histograms Construct a...Ch. 1.4 - Interpreting Relative Frequency Histograms Use the...Ch. 1.4 - Interpreting Relative Frequency Histograms Use the...Ch. 1.4 - Interpreting Relative Frequency Histograms Use the...Ch. 1.4 - Interpreting Relative Frequency Histograms Use the...Ch. 1.4 - Interpreting Relative Frequency Histograms Use the...Ch. 1.4 - Interpreting Relative Frequency Histograms Use the...Ch. 1.4 - Class Boundaries In Exercises 912, use the...Ch. 1.4 - Class Boundaries In Exercises 912, use the...Ch. 1.4 - Prob. 11ECh. 1.4 - Prob. 12ECh. 1.4 - Prob. 13ECh. 1.4 - Prob. 14ECh. 1.4 - Relative Frequency Histogram I Construct a...Ch. 1.4 - Prob. 16ECh. 1.4 - Prob. 17ECh. 1.4 - Prob. 18ECh. 1.4 - Prob. 19ECh. 1.4 - Prob. 20ECh. 1.4 - Prob. 21ECh. 1.4 - Prob. 22ECh. 1.4 - Prob. 23ECh. 1.4 - Prob. 24ECh. 1.4 - A Recurring Illness The length of time (in months)...Ch. 1.4 - Prob. 26ECh. 1.4 - Prob. 27ECh. 1.4 - Prob. 28ECh. 1.4 - Prob. 29ECh. 1.4 - Prob. 30ECh. 1.4 - Prob. 31ECh. 1.4 - Student Heights The self-reported heights of 105...Ch. 1.4 - Prob. 33ECh. 1 - Quantitative or Qualitative? Identify each...Ch. 1 - Symmetric or Skewed? Do you expect the...Ch. 1 - Continuous or Discrete? Identify each variable as...Ch. 1 - Continuous or Discrete, again Identify each...Ch. 1 - World Lakes A lake is a body of water surrounded...Ch. 1 - Prob. 6RWYLCh. 1 - Election Results The 2016 election was a race in...Ch. 1 - Prob. 8RWYLCh. 1 - Prob. 9RWYLCh. 1 - Pulse Rates A group of 50 biomedical students...Ch. 1 - Prob. 11RWYLCh. 1 - Prob. 12RWYLCh. 1 - Gasoline Tax The following are the 2017 state...Ch. 1 - Prob. 14RWYLCh. 1 - Prob. 15RWYLCh. 1 - Prob. 16RWYLCh. 1 - Kentucky Derby The following data set shows the...Ch. 1 - Prob. 18RWYLCh. 1 - Old Faithful The following data are the waiting...Ch. 1 - Prob. 20RWYLCh. 1 - Prob. 1CSCh. 1 - Prob. 2CS
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