Problem #3: Let X denote the proportion of allotted time that a randomly selected student spends working on a certain aptitudetest. Suppose the pdf of X isProblem #3(a):Problem #3(b):f(x; 0) =S(0+1)x000≤x≤1otherwisewhere -1. A random sample of ten students yields the data0.82, 0.55, 0.70, 0.64, 0.45, 0.95, 0.77, 0.84, 0.76, 0.60(a) Use the method of moments to obtain an estimator for 8, and then use it to compute an estimate from the data.(b) Find the maximum likelihood estimator for 8, and then use it to compute an estimate for 8 from the data.answer correct to 2 decimalsanswer correct to 2 decimals

Given information: fx=θ+1xθ, 0≤x≤1 a) Method of moment estimator: Let us find E(X), EX=∫01xfxdx…

Answered: Problem #3: Let X denote the proportion…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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The variance of temperatures (Fahrenheit) in Buffalo, N.Y. is 362.18. What is this variance if temp is re-expressed in Celsius? [Hint: Conversion between Fahrenheit and Celsius is a linear transformation: Fahrenheit = Celsius*1.80 + 32, or Celsius = Fahrenheit*(1/1.80) - (32/1.80)]
Problem#3: The number of messages sent per hour over a computer network has the following distribution: x = number of messages 10 11 12 13 14 15 f(x) 0.08 0.15 0.30 0.20 0.20 0.07 Determine the mean and standard deviation of the number of messages sent per hour.
Problem 4. Suppose that a person's score X on a mathematics aptitude test is a number between 0 and 1, and that their score Y on a music aptitude test is also a number between 0 and 1. Suppose further that in a certain population, the scores X and Y are distributed according to the joint PDF (2x + 3y) for 0≤x, y ≤1 f(x, y) = otherwise (A) What proportion of college students obtain a score greater than 0.8 on the math test? (B) If the student's score on the music test is 0.3, what is the probability that their score on the math test will be greater than 0.8? (C) If a student's score on the math test is 0.3, what is the probability that their music test will be greater than 0.8?
A weight-loss program wants to test how well their program is working. The company selects a simple random sample of 51 individual that have been using their program for 15 months. For each individual person, the company records the individual's weight when they started the program 15 months ago as an x-value. The subject's current weight is recorded as a y-value. Therefore, a data point such as (205, 190) would be for a specific person and it would indicate that the individual started the program weighing 205 pounds and currently weighs 190 pounds. In other words, they lost 15 pounds. When the company performed a regression analysis, they found a correlation coefficient of r = 0.707. This clearly shows there is strong correlation, which got the company excited. However, when they showed their data to a statistics professor, the professor pointed out that correlation was not the right tool to show that their program was effective. Correlation will NOT show whether or not there is…
(g) The following question was investigated: If the standard deviation of the mean for the sampling distribution of random samples of size 64 from a large or infinite population is 5, how large must the sample size become if the standard deviation is to be reduced to 0.9. In solving this question, it was determined that n = 1975.308642. Since we cannot talk to a partial person, how many people do we need to sample? (h) Suppose you collect data and want to find P(X < some number) by using the t-distribution. What do we need to assume about the population to make sure we can use the t-distribution? (i) Suppose we want to solve the following: • Find k such that P(x2 < k) = 0.025 where df = 8. Find P(x? < 4.182) where df = 6. What R function would you use in each case to solve these questions? You do not need to actually solve, just write down the R function. An example of an R function is dbinom (this is not the one that you need).
Problem #4: (2 points) Find a 100(1 - a)% confidence interval for 0, given X₁,,X, i.i.d. with pdf f(x0) = 1,0-0.5 0.
A researcher wants to investigate the effects of environmental factors on IQ scores. For an initial study, she takes a sample of 400 people who grew up as the only child. She finds that 48.5% of them have an IQ score over 100. It is known that 50% of the general population has an IQ score exceeding 100. (a) Find the mean of p, where p is the proportion of people with IQ scores over 100 in a random sample of 400 people. (b) Find the standard deviation of p. (c) Compute an approximation for P(p is greater than or equal to 0.485), which is the probability that there will be 48.5% or more individuals with IQ scores over 100 in a random sample of 400. Round answer to 4 decimal places.

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