STAT-2910-91: Midterm Examination Statistical Methods and Applications

Questions

1. The stem-and-leaf display for the birth weight of 50 newborn babies in a hospital is given below:

Calculate Q1, Q3 , Median, and IQR and determine whether the smallest value of this data set is an outlier.

2. A national engineering council conducted a survey of 1,200 civil engineers who had graduated from the University of British Columbia. For these engineers, the mean salary was found to be $78,000 with a standard deviation of $9,000. If the distribution of salaries of these engineers is mound shaped then:

• How many engineers in the sample had salaries between $87,000 and $96,000?
• How many engineers in the sample had salaries less than $69,000?

3. The following probabilities for two events N, W are given:
P(W ∩ N) = 0.12       P(W ∩ Nᶜ) = 0.28
P(Wᶜ ∩ N) = 0.18      P(Wᶜ ∩ Nᶜ) = 0.42

•  Find P(N ∪ W).
• Find P(N | W)
• Are N and W mutually exclusive? Justify
• Are W and N independent? Justify

4. a)  A wildlife researcher finds that 70% of tagged baby turtles successfully navigate to the ocean after hatching. A random sample of 12 baby turtles is observed during this journey.

• Find the probability that at least 4 turtles do not successfully reach the ocean.
• Find the mean and variance of the number of turtles among the 12 selected that successfully reach the ocean.

b) The number of meteor fragments detected by a ground sensor during a 1-minute interval follows a Poisson distribution. If the probability that no fragments are detected in a minute is 0.01, what is the probability that more than 1 fragment is detected in that minute? 

5. a) Find P(−1.25 < Z xss=removed>

b) A random variable X is normally distributed with unknown mean µ=14 and variance σ2 . If P(X<10>

c) Suppose that a random variable X has mean 10 and standard deviation of 2. A sample of size n=100 is taken from the population represented by this random variable. Compute the ????(????̅ > 12)

6. A public health researcher wants to estimate the proportion of adults in Windsor who regularly exercise using a 90% confidence interval with a total width no larger than 0.06. From previous studies, she believes the true proportion is approximately 0.45. What sample size ???? should be selected to achieve the desired precision?

7. A bottling company wants to estimate the proportion of defective glass bottles produced in a day from its main assembly line. A random sample of 150 bottles is inspected, and 18 are found to be defective

a) Find a 90% confidence interval for the true proportion of defective bottles produced by this assembly line. Interpret this confidence interval. 

b)  The company claims that no more than 5% of its bottles are defective. Based on the confidence interval found in part (a), can one conclude that the true proportion of defective bottles exceeds 5%? Justify your answer.  

c) Perform a formal hypothesis test at significance level α = 0.10 using the p-value method to test whether the true proportion of defective bottles is greater than 0.05. Clearly state the null and alternative hypotheses, compute the test statistic and p-value, and state your conclusion in context. Is this conclusion the same as in part (b)?

8. A study was conducted to investigate the mean number of police emergency calls per 8-hour shift in a particular district of Toronto. A random sample of 100 8-hour shifts was selected from the police records, and the number of emergency calls was recorded for each shift. The following sample statistics were obtained: ???? = 100; ????ˉ = 2.8; ???? 2 = 1.64

a) Find a 98% confidence interval for the mean number of police emergency calls per shift in this district. Interpret the interval.

b) The police chief claims that, on average, there are 3.5 emergency calls per 8-hour shift in this district. Based on the confidence interval in part (a), does the interval suggest that the true mean number of emergency calls per shift is different from 3.5? Explain why or why not. 

c) Perform an appropriate hypothesis test to formally assess whether the mean number of emergency calls per 8-hour shift differs from 3.5, using a suitable significance level and clearly stating: the null and alternative hypotheses; the test statistic and p-value; your conclusion in context.

Summary of Assessment Requirements

This assessment required the student to demonstrate competency across core areas of statistical analysis by solving a set of eight multi-part quantitative questions. The tasks involved:

Key Requirements

• Descriptive Statistics:
  Calculating quartiles (Q1, Q3), median, IQR, and determining outliers using the stem-and-leaf dataset.
• Empirical Rule Applications:
  Using mean, standard deviation, and normal distribution properties to estimate frequencies within given salary ranges.
• Probability Concepts:
  Computing joint, marginal, conditional probabilities; determining whether events are mutually exclusive or independent.
• Binomial & Poisson Distribution Problems:
  Calculating probabilities, mean, and variance for binomial scenarios, and probability outcomes for Poisson-based events.
• Standard Normal (Z) Calculations:
  Computing probabilities involving Z-scores and identifying symmetric probability bounds.
• Normal Distribution Parameter Estimation:
  Using given cumulative probabilities to determine variance and compute sampling distribution probabilities.
• Confidence Intervals & Sample Size Calculation:
  Determining required sample size for given precision, constructing proportion-based confidence intervals, and interpreting them.
• Hypothesis Testing (Proportion & Mean):
  Performing formal hypothesis tests using test statistics and p-values, stating hypotheses clearly, and interpreting results contextually.

Overall, the student had to demonstrate mastery in probability, distributions, estimation, hypothesis testing, and interpretation.

How the Academic Mentor Guided the Student – Step-by-Step Approach

The mentor supported the student by breaking down each section into a clear, logical workflow, reinforcing both conceptual understanding and correct method application.

Understanding the Dataset & Descriptive Statistics

Mentor’s Guidance:

• Reviewed the stem-and-leaf display with the student.
• Demonstrated how to reconstruct the dataset from the stem values.
• Explained the ordered data process for identifying Q1, Q3, and the median.
• Guided the calculation of IQR and the outlier rule (1.5 × IQR).

Outcome: The student gained a strong understanding of data summarisation and identifying unusual values using statistical thresholds.

Applying the Empirical Rule

Mentor’s Guidance:

• Reinforced the concept of mound-shaped (approximately normal) distributions.
• Explained how the 68–95–99.7 rule applies to salary data.
• Helped compute the proportion of values within the given ranges and translate those proportions into counts.

Outcome: The student learned how approximations can be used for real-world inferences using normal distributions.

Probability of Events (N and W)

Mentor’s Guidance:

• Helped the student visualise data using a 2 × 2 probability table.
• Explained how to calculate:
  • P(N ∪ W) using addition rule
  • Conditional probability P(N | W)
  • Tests for mutual exclusivity and independence

Outcome: The student understood relationships between events and how to interpret probability structures rigorously.

Binomial and Poisson Problems

Mentor’s Guidance:

• Demonstrated binomial formulas for “at least” probabilities.
• Guided use of complement rule for simplifying calculations.
• Clarified how to compute binomial mean (np) and variance (npq).
• For Poisson, helped derive λ from P(X = 0) and compute P(X > 1).

Outcome: The student strengthened skills in probability distribution modelling and interpreting real-life random events.

Standard Normal Probability Calculations

Mentor’s Guidance:

• Explained normal curve symmetry and Z-tables.
• Guided computations for P(−1.25 < Z>
• Showed how to find z₀ from the middle probability.
• Supported derivation of variance from given cumulative probability.

Outcome: The student mastered Z-score applications and inverse-normal reasoning.

Sample Size for Desired Precision

Mentor’s Guidance:

• Reinforced the formula for sample size in proportion estimates.
• Explained margin of error, confidence level, and z-values.

Outcome: The student understood study design principles and precision planning.

Confidence Interval and Hypothesis Testing for Proportion

Mentor’s Guidance:

• Reviewed formula for CI of population proportion.
• Explained interpretation in plain language.
• Guided the student through:
  • Writing hypotheses
  • Calculating the test statistic
  • Finding the p-value
  • Stating conclusions clearly
• Emphasised how CI results relate to hypothesis test outcomes.

Outcome: The student became confident in analysing categorical data and deriving data-driven conclusions.

Confidence Interval and Hypothesis Test for Mean

Mentor’s Guidance:

• Reviewed CI formula for population mean with known sample variance.
• Explained interpretation of the 98% CI.
• Supported the hypothesis test step-by-step:
  • Setting hypotheses
  • Calculating test statistic
  • Using p-value to conclude
• Reinforced how CI and hypothesis tests often lead to consistent interpretations.

Outcome: The student strengthened competency in inferential statistics for continuous measurements.

Overall Learning Objectives Achieved

By completing the assessment with mentor guidance, the student demonstrated:

• Ability to summarise, organise, and interpret data using key descriptive statistics.
• Strong understanding of normal, binomial, and Poisson distributions and their applications.
• Proficiency in probability rules, conditional probability, mutual exclusivity, and independence.
• Skilled application of confidence intervals and hypothesis testing for proportions and means.
• Improved capability to interpret statistical results, justify conclusions, and relate findings to real-world contexts.
• Enhanced confidence in structured problem-solving following statistical guidelines.

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