1.On one set of axes, draw the graphs of the pdfs for N(1, 2) and N(-2, 1). For both functions, label the x-axis points corresponding to the probabilities .025, 1/6, .5, 5/6, and .975. (Give the axis values, not the (cdf) probabilities themselves.)
2.On one set of axes, draw the graphs of the pdfs for N(1, 2) and N(-2, 1). For both functions, label the x-axis points corresponding to the values: μ – 2σ, μ – σ, μ, μ + σ, and μ + 2σ. •For a particular distribution, suppose you only have a labeled graph of the cdf (graphed in the normal way – inputs on the x-axis, outputs on the y-axis), and you need to find a quantile. Explain how you would use (which particular axes of) the former to find the desired quantile value. •For a particular distribution, suppose you only have a labeled graph of the quantile function (graphed in the normal way – inputs on the x-axis, outputs on the y-axis), and you need to find a value of the cdf. Explain how you would use the former to find the desired cdf value.
3.The firm you work for has an odd policy that any estimations of the cost of a certain activity must be within a margin of error of no more than ± 10% of the estimated cost. (The company doesn’t care about the confidence level of the margin of error, only that it be such that the above rule is upheld.) A project that you’re working on has an average cost of $10,000, which was derived from a sample of all 11 known observations from a probability distribution with a known standard deviation of 1,869. To comply with your company’s policy, what is the highest confidence level you may use to construct this margin of error?
4.You work in an IT firm that specializes in a certain sort of information retrieval for corporations. Your boss has assigned you the task of estimating how long a certain sort of advanced retrieval takes on average. Since a major business is seriously considering an account with your company, your boss wants you to be “99 percent sure” that your estimate falls within a 10-point total spread that you will give him. The variance of these sorts of retrievals is known to be 4800. What is the least number of trials of the system that you would have to run to complete your task (using the techniques that this test covers)?
9.Suppose you construct a 95% confidence interval for μ, and get the interval (a, b). You later consider the probability distribution of the population of primary interest (intuitively, that of the Xis), and you notice that Pr[a < Xi ≤ b] is much less than 95%. Explain this (using only ideas we have covered in this class).
10.Suppose you take a sample of 25 individuals (from a normally distributed population, when the variance is known) and your confidence interval for the mean at the confidence level .93 is (23, 41). Is the area above the curve from 23 to 41 of the distribution from which you drew these 25 individuals greater than, less than, or equal to .93? Explain your answer (you don’t have to calculate anything, just say why). •Draw and label a picture as part of your explanation of why the innermost c of a normal distribution is the shortest interval of probability c. •Use the argument with differential equations given in class and on the slides to show that the shortest interval of probability c occurs when (and only when) the endpoints of the interval have the same density.
11.Occasionally a standard deviation is presented as a margin of error. If the known standard deviation is 14 from n=20 observations, what is corresponding confidence level c? (I.e., if we assume moe = σ, then what do we know here about c?)
1. You want to use a mean to determine the proportion of women voters in an upcoming election. However, your gender variable G codes men as 13, and women as 17 (those are the only two categories G presents). How would you algebraically transform G so that the resulting mean is the correct proportion?
2. In your random sample of 367 objects, 101 of them were of type A. Construct a point estimate/margin of error (at the 95% confidence level) for the true proportion of objects of type A.
3.What is the 95% confidence interval for the true proportion if there are 229 successes out of 507 trials?
5. Using the T-distribution table, consider the distribution for a confidence interval for the mean from a sample where n = 24.
If Pr [1.83 < X ≤ b] = .03.
What is b?
58. "My 95% confidence interval is (20, 30), so there is a 95% chance that the true mean μ is somewhere between 20 and 30." Explain what is wrong with this claim.
59. Suppose you construct a 95% confidence interval for μ, and get the interval (a, b). You later consider the probability distribution of the population of primary interest (intuitively, that of the Xis), and you notice that Pr[a < Xi≤ b] is much less than 95%. Explain this (using only ideas we have covered in this class).
60. Suppose you take a sample of 25 individuals (from a normally distributed population, when the variance is known) and your confidence interval for the mean at the confidence level .93 is (23, 41). Is the area above the curve from 23 to 41 of the distribution from which you drew these 25 individuals greater than, less than, or equal to .93? Explain your answer (you don’t have to calculate anything, just say why).
The 94% “innermost” interval of T(5) has endpoints ±2.42. What is the probability of this same interval for N(0, 1)?
