Statistics 205 Midterm Exam #1 Review – Step-by-Step Guidance

Q1. What type of sampling problem is present when a pollster surveys 500 subscribers of Alberta Views magazine to estimate the proportion of Canadians supporting new anti-terrorism legislation?

Background

Topic: Sampling Bias in Survey Design

This question tests your understanding of different types of sampling problems, such as selection bias, undercoverage, and non-representative samples.

Key Terms:

• Sampling bias: Systematic error due to a non-random sample of a population.

• Undercoverage: When some members of the population are inadequately represented in the sample.

• Non-representative sample: A sample that does not accurately reflect the characteristics of the population.

Step-by-Step Guidance

1. Identify the target population (all Canadians 18+ years old) and the actual sample (subscribers to Alberta Views magazine).

2. Consider whether the sample is likely to be representative of the target population. Think about the characteristics of magazine subscribers versus the general population.

3. Recall the main types of sampling problems (e.g., selection bias, undercoverage, voluntary response bias) and match the scenario to the most likely issue.

4. Explain why this sampling method could lead to biased results, referencing the definitions above.

Try solving on your own before revealing the answer!

Q2. Probability with Internet Addiction and Gender

Background

Topic: Basic Probability, Joint and Marginal Probabilities, Independence

This question tests your ability to compute probabilities involving two categorical variables and to assess independence.

Key Terms and Formulas:

• Joint probability: $P(A \cap B)$, probability both events occur.

• Marginal probability: Probability of a single event regardless of the other.

• Independence: $P(A \cap B) = P(A)P(B)$

• Complement rule: $P(A^c) = 1 - P(A)$

• Union (not both): $P(A \cup B) - P(A \cap B)$

Step-by-Step Guidance

1. List all given probabilities: $P(\text{male}) = 0.52$, $P(\text{female}) = 0.48$, $P(\text{addict}) = 0.20$, $P(\text{male} \cap \text{addict}) = 0.13$.

2. For (a), use the complement and joint probability rules to find $P(\text{female} \cap \text{not addict})$.

3. For (b), use the formula for the probability of the union of two events, adjusting for "not both" as specified.

4. For (c), recall the definition of independence and check if $P(\text{male} \cap \text{addict}) = P(\text{male}) \times P(\text{addict})$.

Try solving on your own before revealing the answer!

Q3. Experimental vs. Observational Study: Violent Cartoons and Behavior

Background

Topic: Experimental Design, Types of Variables, Randomization

This question tests your understanding of experimental vs. observational studies, control/treatment groups, variable types, and the purpose of randomization.

Key Terms:

• Experimental study: Researcher manipulates variables and assigns subjects to groups.

• Observational study: Researcher observes without intervention.

• Control group: Group not receiving the treatment.

• Treatment group: Group receiving the intervention.

• Randomization: Assigning subjects to groups by chance to reduce bias.

• Categorical variable: Describes qualities or categories.

• Numerical variable: Quantitative, measured with numbers.

Step-by-Step Guidance

1. Determine if the study involves manipulation and random assignment (hallmarks of an experiment).

2. Identify which group is the control and which is the treatment based on the intervention.

3. Consider what outcome would indicate a treatment effect (difference in violent actions).

4. Classify the variable being measured (number of violent actions) as categorical or numerical.

5. Explain why random assignment is important in this context.

Try solving on your own before revealing the answer!

Q4. Diagnostic Test: Sensitivity, Specificity, and Probabilities

Background

Topic: Conditional Probability, Sensitivity, Specificity, Bayes' Theorem

This question tests your ability to use sensitivity, specificity, and prevalence to compute probabilities related to diagnostic tests.

Key Terms and Formulas:

• Sensitivity: $P(\text{Test} + | \text{Disease})$

• Specificity: $P(\text{Test} - | \text{No Disease})$

• Prevalence: $P(\text{Disease})$

• Bayes' Theorem: $P(A|B) = \frac{P(B|A)P(A)}{P(B)}$

• Total Probability: $P(\text{Test} +) = P(\text{Test} + | \text{Disease})P(\text{Disease}) + P(\text{Test} + | \text{No Disease})P(\text{No Disease})$

Step-by-Step Guidance

1. Write down all given probabilities: prevalence, sensitivity, specificity.

2. For (a), use the law of total probability to find $P(\text{Test} +)$.

3. For (b), use Bayes' theorem to find $P(\text{Disease} | \text{Test} +)$.

4. For (c), use Bayes' theorem to find $P(\text{Disease} | \text{Test} -)$.

5. For (d), interpret the probability from (c) in the context of the test and disease.

Try solving on your own before revealing the answer!

Q5. What type of probability has AHS computed in their media release about flu patients?

Background

Topic: Types of Probability (Empirical, Theoretical, Subjective)

This question tests your understanding of the different interpretations of probability.

Key Terms:

• Empirical probability: Based on observed data or experiments.

• Theoretical probability: Based on known possible outcomes.

• Subjective probability: Based on personal judgment or belief.

Step-by-Step Guidance

1. Identify how the probability was calculated (from observed data or theoretical reasoning).

2. Recall the definitions of empirical, theoretical, and subjective probability.

3. Match the scenario to the correct type of probability and explain your reasoning.

Try solving on your own before revealing the answer!

Q6. Relative Risk of Breast Cancer Based on Age at First Birth

Background

Topic: Relative Risk, Risk Interpretation

This question tests your ability to compute and interpret relative risk from two groups.

Key Terms and Formula:

• Relative risk: $RR = \frac{\text{Risk in exposed group}}{\text{Risk in unexposed group}}$

• Risk: $\text{Risk} = \frac{\text{Number of cases}}{\text{Total in group}}$

Step-by-Step Guidance

1. Calculate the risk of breast cancer for each group (before 25, 25 or older).

2. Compute the relative risk using the formula above.

3. Interpret what the relative risk means in the context of the data (higher, lower, or equal risk).

Try solving on your own before revealing the answer!

Q7. Sampling Method Used by the Airline

Background

Topic: Sampling Methods

This question tests your knowledge of the four main random sampling methods: simple random, stratified, cluster, and systematic sampling.

Key Terms:

• Simple random sampling

• Stratified sampling

• Cluster sampling

• Systematic sampling

Step-by-Step Guidance

1. Review the description of the sampling process (one flight per day for seven days).

2. Recall the definitions of each sampling method.

3. Match the scenario to the most appropriate sampling method and explain your reasoning.

Try solving on your own before revealing the answer!

Q8. Difference Between Response Bias and Non-Response Bias

Background

Topic: Survey Biases

This question tests your understanding of different types of bias in survey data collection.

Key Terms:

• Response bias: Systematic pattern of inaccurate answers.

• Non-response bias: Bias introduced when a significant portion of those sampled do not respond.

Step-by-Step Guidance

1. Define response bias and non-response bias.

2. Explain how each type of bias can affect survey results.

3. Provide an example or scenario for each type of bias.

Try solving on your own before revealing the answer!

Q9. Poll of Calgarians on Olympic Bid: Statistic or Parameter, Margin of Error

Background

Topic: Statistics vs. Parameters, Margin of Error, Inference

This question tests your understanding of the difference between sample statistics and population parameters, and how to compute and interpret margin of error.

Key Terms and Formulas:

• Statistic: A value calculated from sample data.

• Parameter: A value that describes a population.

• Margin of error (for proportions): $ME = z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$ (commonly $z^* = 2$ for 95% confidence)

Step-by-Step Guidance

1. Determine whether the value 0.3498 is a statistic or a parameter and justify your answer.

2. Use the margin of error formula for proportions, plugging in the sample proportion and sample size.

3. Interpret what the margin of error means in the context of the poll.

4. Consider whether the sample result allows you to infer the population proportion is 30%, and explain why or why not.

Try solving on your own before revealing the answer!

Q10. Telemarketing Calls: Median, Variance, and Standard Deviation

Background

Topic: Descriptive Statistics (Measures of Center and Spread)

This question tests your ability to compute and interpret the median, variance, and standard deviation for a small data set.

Key Terms and Formulas:

• Median: Middle value when data are ordered.

• Variance (sample): $s^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2$

• Standard deviation: $s = \sqrt{s^2}$

Step-by-Step Guidance

1. Order the data from smallest to largest to find the median.

2. Interpret the median in the context of the data (number of calls).

3. Compute the variance using the formula above, with the given mean.

4. Compute the standard deviation as the square root of the variance.

Try solving on your own before revealing the answer!