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2B03 Assignment 3

Sampling Distributions and Statistical Inference

Instructions: You are to use R Markdown for generating your assignment (see the item Assignments and R Markdown on the course website for helpful tips and pointers).

  1. Define the following terms in a sentence (or short paragraph) and state a formula if appropriate (this question is worth 5 marks).
    1. Sampling Distribution
    2. Estimate
    3. Interval Estimate
    4. Bias
    5. Hypothesis Test
  2. Suppose that the number of hours per week of lost work due to illness in a certain automobile assembly plant is approximately normally distributed, with a mean of 45 hours and a standard deviation of 15 hours. For a given week, selected at random, what is the probability that (this question is worth 3 marks):
    1. The number of lost work hours will exceed 75 hours?
    2. The number of lost work hours will be between 35 and 45 hours?
    3. The number of lost work hours will be exactly 45 hours?
  3. If the income in a community is normally distributed, with a mean of $39,000 and a standard deviation of $8,000, what minimum income does a member of the community have to earn in order to be in the top 5%? What is the minimum income one can have and still be in the middle 50% (this question is worth 4 marks)?
  4. A senator claims that 60% of her constituents favor her voting policies over the past year. In a random sample of 50 of these people, the sample proportion of those who favored her voting policies was only 0.5. Is this enough evidence to make the senator’s claims strongly suspect? (Hint: Use a normal approximation to the binomial distribution then construct a confidence interval – this question is worth 2 marks).
  5. A cereal company checks the weight of its breakfast cereal by randomly checking 62 of the boxes. This particular brand is packed in 20-ounce boxes. Suppose that a particular random sample of 62 boxes results in mean weight of 20.02 ounces. How often will the sample mean be this high, or higher if μ = 20 and σ = 0.10 (this question is worth 4 marks)?
  6. I wish to estimate the proportion of defectives in a large production lot with plus or minus D = 0.05 of the true proportion, with a 90% level of confidence. From past experience, it is believed that the true proportion of defectives is π = 0.02. How large a sample must be used? (Hint: Use a normal approximation for the sample proportion Pˆ – this question is worth 2 marks).