examples
Page 1: Appearance Defects in New Cars
Consumer Organization Findings:
Many new cars had appearance defects (dents, scratches, paint chips).
None had more than three defects.
Prevalence of defects:
7% had three defects
11% had two defects
21% had one defect
a) Find the expected number of defects:
Let X be the number of defects.
E(X) = (0 * P(0)) + (1 * P(1)) + (2 * P(2)) + (3 * P(3)).
b) Calculate the standard deviation:
Use the formula: SD = sqrt(E(X^2) - (E(X))^2).
Page 2: Probability of Contracts
Scenario: Two contract bids with different probabilities.
a) Independence of Outcomes:
Are contract bids independent?
b) Both Contracts Probability:
P(get Contract #1 and #2) = P(Contract #1) * P(Contract #2 | Contract #1).
c) Probability of Neither Contract:
P(neither #1 nor #2) = 1 - P(Contract #1 or Contract #2).
d) Expected Value (X):
Define X based on the possible outcomes and find E(X) and SD.
Page 3: Broken Eggs in Purchase
Assumptions about Broken Eggs:
Mean broken eggs per dozen = 0.6; SD = 0.5.
a) Expected broken eggs:
E(broken eggs in 3 dozen) = 3 * 0.6.
b) Standard deviation of broken eggs:
SD(broken eggs in 3 dozen) = sqrt(3) * 0.5.
c) Independence Assumption:
Evaluate if independence of cartons is necessary and why it matters.
Page 4: Insurance Company Profit Estimation
Estimates: Annual profit of $150 with SD = $6000.
a) Discussing large SD:
Explore reasons behind a high standard deviation in profit estimates.
b) Profit from Two Policies:
Mean and SD is given by: Mean = 2 * $150; SD = SD(1st) + SD(2nd).
c) Profit from 1000 Policies:
Calculate mean and standard deviation for 1000 policies.
d) Independence Violation Circumstances:
Discuss scenarios where independence assumption can break down.
Page 5: Bicycle Shop Profit Estimation
Children's Bicycle Sales:
Basic model profit = $120; Deluxe model = $150.
Define Random Variables:
Let X1 = sales of basic model; X2 = sales of deluxe model.
Net profit = 120X1 + 150X2 - 200.
b) Mean Net Profit:
Calculate expected value based on mean sales.
c) Standard Deviation of Net Profit:
Apply rules for variance of independent random variables.
d) Assumptions for Mean and Standard Deviation:
Discuss necessary assumptions for averages and variability calculations.
Page 6: Random Variable Type
Dropped Calls (Y):
Identify random variable Y as:
A) measurable
B) discrete
C) continuous
D) a failure.
Page 7: Expected Drops Calculation
Values of Y: 0, 1, 2, 3 with probabilities 0.55, 0.30, 0.10, 0.05.
Find expected dropped calls per day:
E(Y) = sum(values * probabilities).
Page 8: Probability Model for Calls
Probability of a Dropped Call:
If p = 0.02, determine the appropriate model for dropped calls:
A. Binomial
B. Uniform
C. Geometric
D. Poisson.
Page 9: Random Variable Time
Technical Support Completion Time (Y):
Identify Y as:
A) discrete
B) continuous
C) binomial
D) expected.
Page 10: Normal Probability Model Characteristics
Identify Incorrect Statement:
Find which is NOT a characteristic of the normal model:
A. Normal variable is discrete
B. Bell-shaped distribution
C. Sum of independent normals is also normal
D. Mean equals median equals mode.
Page 11: Study Time Distribution
Study Hours Distribution:
Typically, study time for students has mean = 6 hours; SD = 1.5.
More than 8 Hours Probability:
Find corresponding probability based on normal curve calculations.
Page 12: Normal Approximation Conditions
Conditions for Binomial Approximation:
Must meet these criteria:
A. Two outcomes per trial
B. Trials independence
C. Expected successes and failures ≥ 10.
Page 13: Sampling Distribution of Smartphone Users
Proportion of Smartphone Users:
Sample size = 200; true proportion = 36%.
Expected Distribution Shape:
Likely to be approximately normal due to sample size.
Mean and Standard Deviation:
Find mean and SD of the sampling distribution.
Page 14: Women in Random Sample
Adult Women Proportion:
Approximately 50% in US; survey of 400 people.
Surprises:
Assessing likelihood of finding different percentages in the sample.
Page 15: Contact Lenses Proportion
Percentage Wearing Contacts:
True proportion = 30%; random sample of 100.
Probability more than one-third wear contacts:
Calculate likelihood based on proportions.
Page 16: Confidence Interval for Teen Driver Accidents
Teenager Involvement in Accidents:
582 accidents analyzed; confidence interval establishment.
Interpretation:
Define and explain 95% confidence intervals for estimates.
Page 17: Flu Vaccine Confidence Interval
Flu Vaccine Proportion:
48% of 200 surveyed; CI = (0.409, 0.551).
Sample Size Impact:
Discuss how sample sizes or confidence levels affect CI.
Page 18: Margin of Error
Market Research Margin Theory:
Identify significance of margin of error in estimation.
Reducing Margin:
Ways to decrease margin of error in surveys.
Page 19: Sample Size for Emission Standards
Sample Size Calculation:
Based on findings of cars with faulty emissions and required margin of error.
Page 20: Customer Satisfaction with a Restaurant
Satisfaction Probability Examination:
80% satisfaction claim; random sample of 9.
Calculate various probability outcomes.
Page 21: Chip Manufacturing Rejection Rates
Chances for Rejected Chips:
Test various scenarios of rejection rates in random samples.
Page 22: E-commerce Purchase Rates
Probability of Purchases:
Analyze probabilities of purchase outcomes from sampled visits.
Page 23: Battery Reliability Rate
Probability for Functional Batteries:
Work with reliable rates in sampling.
Page 24: Coin Toss Probability
Using Binomial Model:
Calculate probabilities for outcomes of tossing coins.
Page 25: Hypothesis Definitions
Define and differentiate between null and alternative hypotheses in various contexts:
a) Proportion change in reports.
b) Enrollment in wellness class.
c) Political majority in polls.
Page 26: Warranty Problems in Computer Model
Explore null and alternative hypotheses regarding warranty claims evaluation.
Page 27: Credit Union Participation Survey
Employee Participation Hypothesis Examination:
Analyze evidence supporting or refuting participation levels.
Page 28: p-value Interpretations
Evaluate statements about significance and null hypothesis validity.
Page 29: Anti-Drug Effectiveness Testing
Testing New Formula: p-value interpretation and conclusions.
Page 30: Mutual Fund Performance Testing
Hypothesis Testing: Evaluating manager's performance against claims.
Page 31: Sample Results Interpretation
Evaluate correct conclusions regarding probability and sample results.
Page 32: Webzine Subscription Interest
Hypothesis Testing for launch decision based on survey results.
Page 33: Medicare Payments Audit
Considerations for hypothesis testing with Medicare payments statistics.
Page 34: Printer Brand Recognition Advertising Decision
Evaluate whether to continue advertising based on public recognition results.
Page 35: Customer Satisfaction Complaint Rate
Statistical analysis of complaints rate and formulation of hypotheses testing.
Page 36: Assembly Line Monitoring
Quality Control Hypotheses: Errors consideration and implications in assembly.
Page 37: Average Age of Online Consumers
Evaluate demographics against historical averages through hypothesis testing.
Page 38: Fuel Economy Goal Examination
Check average MPG against operational goals with hypothesis testing.
Page 39: Hypothesis Testing and Assumptions
Investigate hypotheses for car fleet mileage evaluations.
Page 40: Battery Lifespan Testing Against Claims
Evaluate company’s claims regarding battery longevity through sampling.
Page 41: Sample Size Calculation for GPA Estimation
Determine sample size needed for confidence in estimating student GPA.
Page 42: Health Care Cost Audit
Analysis of out-of-pocket cost distributions among employees through sampling.
Page 43: Valentine’s Day Spending Analysis
Investigate spending of men around Valentine's through sampling statistics.
Page 44: Foreclosed Home Sales Analysis
Conduct confidence interval checks for averages and interpret results.
Page 45: Lunch Cost Interpretation
Discuss various interpretations on CI for lunch spending calculations.
Page 46: Battery Longevity Analysis with Outlier Management
Address data validity and outlier considerations in battery duration testing.