MATH1180 Exam 3: Normal Distribution

• This is MATH1180 Exam 3, Version B

Instructions

• Print your name on the line provided.
• Answer each question fully within the examination booklet.
• Show all work and simplify each answer.
• Full credit requires showing your work, unless stated otherwise.
• The final answer is less important than the reasoning.
• Correct answers without supporting work receive little to no credit.
• Do not open the booklet until instructed.

Problem 1B

• X is a normally distributed random variable with a mean (µ) of 6.
• The standard deviation (σ) is not provided.
• Determine P(X ≤ 6).
• Given P(4.2 ≤ X ≤ 7.8) = 0.997, calculate P(X ≤ 4.2) and find how many standard deviations away from the mean is 4.2.
• Find the value of the standard deviation σ, based on Part b.
• Calculate P(5.5 ≤ X ≤ 7).

Problem 2B

• VO2 max measures the maximum amount of oxygen used during intense exercise.
• VO2 max is know as "the 401K of longevity".
• VO2 max is considered the best indicator of cardiovascular fitness and aerobic endurance,
• The VO2 max average for people aged 20-29 is 2.82 liters/min, with a standard deviation of 0.76 liters/min.
• A study uses a simple random sample of 64 people aged 20-29.
• You must construct and interpret the 95% central interval for the average VO2 max of these 64 people.
• Interpret the 95% central interval for the average VO2 max of these 64 people.
• Determine if the central interval will be longer or shorter for each change: Decreasing sample size, Increasing 1 − α, Increasing sample size, Increasing α

Problem 3B

• In a study on diet and blood pressure, systolic blood pressures (mmHg) were measured in a random sample of 260 vegetarian men.
• An approximate 85% confidence interval for mean systolic blood pressure in vegetarian men is [102.44, 103.56].
• Find the sample mean and margin of error.
• Find the sample standard deviation.
• Using the answers from part a and b, find an approximate 95% confidence interval for mean systolic blood pressure explain whether the interval is longer or shorter.

Problem 4B

• The Verbal Development Quotient (VDQ) assesses language development in young children.
• VDQ scores are normally distributed with a mean of 105 and a standard deviation of 12.
• Determine the probability that a randomly selected child has a VDQ score between 100 and 120.
• Find the central interval that contains the middle 85% of the VDQ scores.
• Determine the percentage of children with above-mean VDQ scores who are classified as having "advanced verbal development," defined as a VDQ exceeding 125.