STAT 2910: Department of Mathematics and Statistics Final Examination

Assessment 

Questions

1. (a) One hundred shoppers at a local shopping mall were categorized by age and gender as shown in the frequency distribution below. One shopper is selected at random from that group of 100 shoppers.

• If the randomly selected shopper is male, what is the probability that he is under 25 years of age? 
• What is the probability that the randomly selected shopper is female or over 40 years of age? 
• Are gender of the shopper and age independent events? Explain. 

(b) Approximately 5% of the bolts coming off a production line have serious defects. Two bolts are randomly selected for inspection. What is the probability that one bolt is defective? Both bolts are defective?

2. (a) The distribution of IQ scores for high school graduates is normally distributed with mean 400 and standard deviation 80. 

• What is the 90th percentile of this distribution of the IQ scores?
• Find the IQ scores that form the boundaries of the middle 60% of the distribution of the scores.
• What is the probability that a randomly selected score exceeds 375?

(b) An airline finds that 5% of the persons making reservations on a certain flight will not show up for the flight. If the airline sells 160 tickets for a flight that has only 155 seats, what is the probability that a seat will be available for every person holding a reservation and planning to fly?

3. A student government representative at a local university claims that 60% of the undergraduate students favour a move to Division I in college football. A random sample of 250 undergraduate students was selected and 140 students indicated they favoured a move to Division I.

(a) Find a 95% confidence interval for the true proportion of undergraduate students who favor the move to Division I. 

(b) Based on the interval in the previous question, can the representative’s claim be rejected? Justify your answer.

4. (a) A process control engineer wishes to estimate the true proportion of defective computer chips with a margin of error of no more than 0.09 and with probability 0.90. How many observations does the engineer need to include in the sample to achieve his goal? 

(b) The Postmaster at the Huntington Post Office would like to compare the delivery times to two different locations which are the same distance from Huntington. A random sample of letters are to be divided into two equal groups, the first delivered to Location A and the second delivered to Location B. Each letter will be delivered on a randomly selected day and the number of days for each letter to arrive at its destination is recorded. The measurements for both groups are expected to have a range (variability) of approximately 4 days. If the estimate of the difference in mean delivery times is desired to be correct to within 1 day with probability equal to 0.99, how many letters must be included in each group? 

Brief Summary of Assessment Requirements

This assessment evaluates students’ understanding and application of probability theory, statistical distributions, inferential statistics, and sample size determination. The questions are designed to test both conceptual knowledge and problem-solving ability across real-world statistical scenarios.

Key Assessment Pointers

• Application of conditional probability and independence of events
• Use of binomial probability models
• Interpretation of the normal distribution, percentiles, and probabilities
• Understanding of sampling distributions and confidence intervals
• Hypothesis-based reasoning using confidence intervals
• Determination of required sample sizes for proportions and means under given confidence levels and margins of error
• Logical explanation and statistical justification of results

Academic Mentor’s Step-by-Step Approach

The academic mentor guided the student through the assessment using a systematic, concept-first approach, ensuring clarity at each stage.

Step 1: Interpreting the Problem Statements

• The mentor began by breaking down each question to identify:
  • Given data
  • Required probabilities or estimates
  • Appropriate statistical methods
• Emphasis was placed on understanding what is being asked before performing calculations.

Step 2: Probability and Independence (Question 1)

• The mentor explained conditional probability using frequency distributions.
• Gender and age categories were analyzed to compute probabilities accurately.
• The concept of independent vs. dependent events was clarified using probability rules.
• For defective bolts, the mentor guided the student in applying binomial probability formulas, distinguishing between “exactly one defect” and “both defective.”

Step 3: Normal Distribution Applications (Question 2)

• The mentor reviewed properties of the normal distribution, including mean, standard deviation, and symmetry.
• Standardization using z-scores was demonstrated step by step.
• The student was guided on:
  • Finding percentiles
  • Determining central probability ranges
  • Calculating probabilities exceeding a given value
• In the airline overbooking scenario, the mentor connected real-world decision-making with probability modeling.

Step 4: Confidence Intervals and Claims (Question 3)

• The mentor explained the logic behind confidence intervals for proportions.
• Each step calculating sample proportion, standard error, and margin of error was addressed clearly.
• The student was guided on how to interpret the interval and assess whether the representative’s claim was statistically supported or rejected.

Step 5: Sample Size Determination (Question 4)

• The mentor introduced formulas for determining sample size for:
  • Proportions (using margin of error and confidence level)
  • Differences in means (considering variability and desired accuracy)
• Real-world implications of choosing appropriate sample sizes were discussed to reinforce understanding.

Outcome Achieved

Through guided instruction:

• The student successfully applied statistical formulas and concepts accurately.
• Each question was answered with logical reasoning and proper justification.
• The student developed confidence in interpreting results rather than only performing calculations.

Learning Objectives Covered

• Understanding and application of probability rules
• Analysis of independence and conditional events
• Use of normal distribution in real-world contexts
• Construction and interpretation of confidence intervals
• Statistical decision-making based on sample data
• Determination of appropriate sample sizes
• Clear communication of statistical reasoning

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