# Question 1 (15 Points) Concepts and Terminology: Answer the questions below in 1-2 sentences for each question. a) Provide two reasons why we would want to sample a subset of values from a population...

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Question 1 (15 Points) Concepts and Terminology: Answer the questions below in 1-2 sentences for each question. a) Provide two reasons why we would want to sample a subset of values from a population rather than observe the entire population. b) Assume we sample a small population of size ?? = 50 with replacement after observing each value. Do we model this as a finite or infinite population? Why? c) Describe the difference between the sample mean and the mean of the sample variance. d) What are two traits of a good estimator? e) Consider a general Gaussian random variable, ??, that has mean ??� and variance ????2. How is ????(??) defined in terms of the q function? (Recall that ????(??) is the cdf of ??) Question 2 (20 Points) You want to analyze the average resistance of resistors produced by a company; but you are only able to sample a few of the resistors. The true mean of the population is 10 ohms. The true standard deviation is 3 ohms. For parts (a) through (d) assume that you randomly observe 10 resistors and observe that the sampled resistors have the following resistance: {5Ω, 7Ω, 7Ω, 8Ω, 8Ω, 9Ω, 10Ω, 11Ω, 14Ω, 14Ω} a) Define the population size, sample size, population mean, and population variance b) Determine the sample mean for the observed sample c) Determine the sample variance for the observed sample d) Determine the unbiased estimate of the population variance for the observed sample e) Determine the expected value of the sample mean f) Determine the variance of the sample mean for a sample size of 10 g) Assume you have a package of 100 packets. The mean and standard deviation for resistors in this package are also 10 and 3, respectively. Determine the variance of the sample mean if a sample of 20 resistors is randomly selected from this package (without replacement). h) For the scenario described in (g), determine the sample size required to assure that the standard deviation of the sample mean is at most 0.5. Question 3 (10 Points) Gaussian and Student’s t distributions. For each question, write your answer in terms of ??( ) or ????( ) and then determine the value from the appropriate table. a) Assume you have a standardized Gaussian random variable, ??. Determine the probability that ?? falls between -0.5 and 1. b) Assume you have a Gaussian random variable, ??, with a mean of 1.5 and a standard deviation of 1. Determine the probability that ?? falls between -0.5 and 1. c) Consider the Student’s t distribution with ?? = 8. Determine the value of ?? where ??(−?? ≤ ?? < )="0.9" d)="" consider="" the="" student’s="" t="" distribution="" with="" =="" 10.="" determine="" the="" value="" of="" where="" (??=""> ??) = 0.025. Q function, ??(??) Student’s t Distribution Function, ????�??� at ?? = ?? − ?? Question 4 (20 Points) Confidence Intervals and Hypothesis Testing a) Consider collecting a sample of 16 random values from a population with Gaussian distribution whose mean is 50 and variance is 9. Determine the 80-percent confidence interval for the sample mean. b) Repeat part (a) with the assumption that the population variance is unknown; but assume that you have determined that the unbiased estimate of the variance is 9 for your sample. c) Assume that the mean and variance of the population are unknown; but there is a claim that the population mean is 50. Additionally, assume that you take a sample of size 16 that gives a sample mean of 48.6 and an unbiased estimate of the variance of 9. Should you accept this claim with a 95% confidence level? d) Repeat part (c) for a claim that the population mean is at least 50. Question 5 (15 Points) Curve Fitting and Linear Regression NOTE: You can use Matlab to confirm your answer; but you should show the calculation for full credit. a) Plot the points in a scatter plot and draw an estimate of the linear regression curve. b) Determine the linear regression equation for the measured values in the table above. c) Determine the Linear Correlation Coefficient (i.e., Pearson’s r) for the dataset in the table above. ?? 1 2 3 4 Value 1 (????) 10 13 17 20 Value 2 (????) 2 6 13 17 Question 6 (15 Points) Consider a non-deterministic continuous random process, ??(??), that is stationary and ergodic. The process has a Gaussian distribution with a mean of 1 and a variance of 4. a) Draw and label the pdf and cdf of ??(??) b) Determine the probability that ??(??) > 4 c) Determine the probability that ??(??) ≤ 4 d) Assume that the process described above represents a voltage that is passed into a comparator. The threshold is set to 4V so that ??(??) = 0V when ??(??) ≤ 4 and ??(??) = 5V when ??(??) > 4. Draw the pdf of ??(??). e) Is ??(??) a continuous random variable or a discrete random variable? Provide a reason for your answer.
Apr 14, 2022
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