Answer the following questions using information provided: Let X1 ,.....Xn be n i.i.d random variables from N(μ,σ2) distribution. i) If σ2 is known, derive an exact 100(1-σ)% confidence interval for μ. ii) If σ2 is unknown, derive an exact 100(1-σ)% confidence interval for μ. iii) An electronic manufacturing company wishes to estimate the average weight of laptops by finding a 99% confidence interval (CI). A worker randomly selects 12 laptops and measures the weight of each laptop. He finds the sample mean is x̄ =47.2 and the sample standard deviation S=5.8. Stating your assumptions about the laptop weights, find a 99% C.I. for the average weight.
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
Answer the following questions using information provided:
Let X1 ,.....Xn be n i.i.d random variables from N(μ,σ2) distribution.
i) If σ2 is known, derive an exact 100(1-σ)% confidence interval for μ.
ii) If σ2 is unknown, derive an exact 100(1-σ)% confidence interval for μ.
iii) An electronic manufacturing company wishes to estimate the average weight of laptops by finding a 99% confidence interval (CI). A worker randomly selects 12 laptops and measures the weight of each laptop. He finds the sample mean is x̄ =47.2 and the sample standard deviation S=5.8. Stating your assumptions about the laptop weights, find a 99% C.I. for the average weight.
Note: use R to compute the value of the necessary quantile for constructing the C.
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