Pearson eText for Basic Business Statistics -- Instant Access (Pearson+)
14th Edition
ISBN: 9780137400119
Author: MARK BERENSON, David Levine
Publisher: PEARSON+
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Chapter 5, Problem 10PS
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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
Pearson eText for Basic Business Statistics -- Instant Access (Pearson+)
Ch. 5 - Given the following probability distributions. a....Ch. 5 - The following table contains the probability...Ch. 5 - Recently , a regional automobile dealership sent...Ch. 5 - In the carnival game Under-or -Over-Seven, a pair...Ch. 5 - The number of arrivals per minute at a bank...Ch. 5 - The manager of the commercial mortgage department...Ch. 5 - You are trying to develop a strategy for investing...Ch. 5 - You plan to invest $1,000 in a corporate bond fund...Ch. 5 - Determine the following:...Ch. 5 - Determine the mean and standard deviation of the...
Ch. 5 - The increase or decrease in the price of a stock...Ch. 5 - According to the Pew Research Center, 51 of...Ch. 5 - A student is taking a multiple-choice exam in...Ch. 5 - A manufacturing company regularly conducts quality...Ch. 5 - Past records indicates that the probability of...Ch. 5 - In Example 5.4 on page 209, you and two friends...Ch. 5 - In Example 5.4 on page 209, you and two friends...Ch. 5 - Assume a Poisson distribution....Ch. 5 - Assume a Poisson distribution....Ch. 5 - Assume a Poisson distribution with =5.0. What is...Ch. 5 - Assume that the number of airline customer service...Ch. 5 - The quality control manager of Marilyn’s Cookies...Ch. 5 - Refer to Problem 5.22. How many cookies in a batch...Ch. 5 - Prob. 24PSCh. 5 - The U.S Department of Transportation maintains...Ch. 5 - The Consumer Financial Protection Bureau’s...Ch. 5 - J.D. Power and Associates calculates and publishes...Ch. 5 - Refer to Problem 5.27. if you purchased a Toyota...Ch. 5 - A toll-free phone number is available from 9 A.M....Ch. 5 - Prob. 30PSCh. 5 - What are the four properties that must be present...Ch. 5 - Prob. 32PSCh. 5 - Darwin Head, a 35-year-old sawmill worker, won 1...Ch. 5 - Between 1896-when the Dow Jones index was...Ch. 5 - Smartphone adoption among American younger adults...Ch. 5 - One theory concerning the Dow jones industrial...Ch. 5 - Prob. 37PSCh. 5 - Prob. 38PSCh. 5 - Social log-ins involve recommending of sharing an...Ch. 5 - The Consumer Financial Protection Bureau’s...Ch. 5 - Refer to Problem 5.40. In the same time period,...Ch. 5 - One theory concerning the...Ch. 5 - Spurious correlation refers to the apparent...Ch. 5 - The United Auto Courts Reports blog notes that are...
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- 2. Hypothesis Testing - Two Sample Means A nutritionist is investigating the effect of two different diet programs, A and B, on weight loss. Two independent samples of adults were randomly assigned to each diet for 12 weeks. The weight losses (in kg) are normally distributed. Sample A: n = 35, 4.8, s = 1.2 Sample B: n=40, 4.3, 8 = 1.0 Questions: a) State the null and alternative hypotheses to test whether there is a significant difference in mean weight loss between the two diet programs. b) Perform a hypothesis test at the 5% significance level and interpret the result. c) Compute a 95% confidence interval for the difference in means and interpret it. d) Discuss assumptions of this test and explain how violations of these assumptions could impact the results.arrow_forward1. Sampling Distribution and the Central Limit Theorem A company produces batteries with a mean lifetime of 300 hours and a standard deviation of 50 hours. The lifetimes are not normally distributed—they are right-skewed due to some batteries lasting unusually long. Suppose a quality control analyst selects a random sample of 64 batteries from a large production batch. Questions: a) Explain whether the distribution of sample means will be approximately normal. Justify your answer using the Central Limit Theorem. b) Compute the mean and standard deviation of the sampling distribution of the sample mean. c) What is the probability that the sample mean lifetime of the 64 batteries exceeds 310 hours? d) Discuss how the sample size affects the shape and variability of the sampling distribution.arrow_forwardA biologist is investigating the effect of potential plant hormones by treating 20 stem segments. At the end of the observation period he computes the following length averages: Compound X = 1.18 Compound Y = 1.17 Based on these mean values he concludes that there are no treatment differences. 1) Are you satisfied with his conclusion? Why or why not? 2) If he asked you for help in analyzing these data, what statistical method would you suggest that he use to come to a meaningful conclusion about his data and why? 3) Are there any other questions you would ask him regarding his experiment, data collection, and analysis methods?arrow_forward
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