EBK STATISTICS FOR BUSINESS & ECONOMICS
13th Edition
ISBN: 8220101456380
Author: Anderson
Publisher: CENGAGE L
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Chapter 18.1, Problem 5E
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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 18 Solutions
EBK STATISTICS FOR BUSINESS & ECONOMICS
Ch. 18.1 - The following hypothesis test is to be conducted....Ch. 18.1 - Ten individuals participated in a taste test...Ch. 18.1 - The median number of part-time employees at...Ch. 18.1 - Net assets for the 50 largest stock mutual funds...Ch. 18.1 - The median price of homes in Austin, Texas is...Ch. 18.1 - The median annual income for families living in...Ch. 18.1 - Prob. 7ECh. 18.1 - A Pew Research Center survey asked adults if their...Ch. 18.1 - In a recent poll six hundred adults were asked a...Ch. 18.1 - Nielsen Media Research identified Empire and the...
Ch. 18.1 - Competition in the personal computer market is...Ch. 18.2 - Two fuel additives are tested to determine their...Ch. 18.2 - A sample of 10 men was used in a study to test the...Ch. 18.2 - Percents of on-time arrivals for flights in the...Ch. 18.2 - A test was conducted for two overnight mail...Ch. 18.2 - The LPGA ANA Inspiration tournament was held in...Ch. 18.2 - The Scholastic Aptitude Test (SAT) consists of...Ch. 18.3 - Two fuel additives are being tested to determine...Ch. 18.3 - Samples of starting annual salaries for...Ch. 18.3 - The gap between the earnings of men and women with...Ch. 18.3 - Are Southern and Western states equally prone to...Ch. 18.3 - Each year Bloomberg Businessweek publishes...Ch. 18.3 - Police records show the following numbers of daily...Ch. 18.3 - A certain brand of microwave oven was priced at 10...Ch. 18.3 - Chicago Midway International Airport and...Ch. 18.4 - A sample of 15 consumers provided the following...Ch. 18.4 - Three admission test preparation programs are...Ch. 18.4 - Forty-minute workouts of one of the following...Ch. 18.4 - The National Football League (NFL) holds its...Ch. 18.4 - A large corporation sends many of its first-level...Ch. 18.4 - The better-selling candies are often high in...Ch. 18.5 - Consider the following set of rankings for a...Ch. 18.5 - Prob. 33ECh. 18.5 - Prob. 34ECh. 18.5 - A national study by Harris Interactive, Inc.,...Ch. 18.5 - Prob. 36ECh. 18.5 - A student organization surveyed both current...Ch. 18 - A survey asked the following question: Do you...Ch. 18 - Due to a recent decline in the housing market, the...Ch. 18 - Twelve homemakers were asked to estimate the...Ch. 18 - Prob. 41SECh. 18 - The following data are product weights for the...Ch. 18 - A client wants to determine whether there is a...Ch. 18 - Prob. 44SECh. 18 - Prob. 45SECh. 18 - Prob. 46SECh. 18 - Prob. 47SE
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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
- Businessarrow_forwardWhat is the solution and answer to question?arrow_forwardTo: [Boss's Name] From: Nathaniel D Sain Date: 4/5/2025 Subject: Decision Analysis for Business Scenario Introduction to the Business Scenario Our delivery services business has been experiencing steady growth, leading to an increased demand for faster and more efficient deliveries. To meet this demand, we must decide on the best strategy to expand our fleet. The three possible alternatives under consideration are purchasing new delivery vehicles, leasing vehicles, or partnering with third-party drivers. The decision must account for various external factors, including fuel price fluctuations, demand stability, and competition growth, which we categorize as the states of nature. Each alternative presents unique advantages and challenges, and our goal is to select the most viable option using a structured decision-making approach. Alternatives and States of Nature The three alternatives for fleet expansion were chosen based on their cost implications, operational efficiency, and…arrow_forward
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