Questions and Answers of Introduction To Probability Statistics

A random sample X1, X2, X3, . . ., X100 is given from a distribution with known variance Var(Xi) = 81. For the observed sample, the sample mean is X̅ = 50.1. Find an approximate 95% confidence
For the following examples, find the maximum likelihood estimator (MLE) of θ:1. Xi ∼ Binomial(m, θ), and we have observed X1, X2, X3, . . ., Xn. 2. Xi ∼ Exponential(θ) and we have observed
In this problem, we would like to find the PDFs of order statistics. Let X1,…,Xn be a random sample from a continuous distribution with CDF FX(x) and PDF fX(x). Define X(1),…,X(n) as the order
To estimate the portion of voters who plan to vote for Candidate A in an election, a random sample of size n from the voters is chosen. The sampling is done with replacement. Let θ be the portion of
Let Z ∼ N(0, 1), find xl and xh such that P ( ₁₁ ≤ 2 ≤ 2₂ ) = 0 xh = 0.95
Let X1, X2, X3, . . ., X100 be a random sample from a distribution with unknown variance Var(Xi) = σ2
Let X1, X2, X3, . . ., Xn be a random sample from a normal distribution N(θ, 1). Find a 95% confidence interval for θ.
A random sample X1, X2, X3, . . ., X36 is given from a normal distribution with unknown mean μ = EXi and unknown variance Var(Xi) = σ2. For the observed sample, the sample mean is X̅ = 35.8, and
Check that the random variables Q1 = X̅ −θ and Q2 = √n(X̅ − θ) are both valid pivots in Example 8.13.Example 8.13:Let X1, X2, X3, . . ., Xn be a random sample from a normal distribution
Let X1, X2, X3, X4, X5 be a random sample from a N(μ, 1) distribution, where μ is unknown. Suppose that we have observed the following values 5.45, 4.23, 7.22, 6.94, 5.98We would like to decide
Let X1, X2, X3, . . ., Xn be a random sample from a distribution with known variance Var(Xi) = σ2, and unknown mean EXi = θ. Find a (1 −α) confidence interval for θ. Assume that n is large.
Let X1,…,X9 be a random sample from a N(μ, 1) distribution, where μ is unknown. Suppose that we have observed the following values 16.34, 18.57, 18.22, 16.94, 15.98, 15.23, 17.22, 16.54, 17.54We
An engineer is measuring a quantity θ. It is assumed that there is a random error in each measurement, so the engineer will take n measurements and report the average of the measurements as the
Let X1, X2 ,..., X150 be a random sample from an unknown distribution. After observing this sample, the sample mean and the sample variance are calculated to beDesign a level 0.05 test to choose
We would like to estimate the portion of people who plan to vote for Candidate A in an upcoming election. It is assumed that the number of voters is large, and θ is the portion of voters who plan to
Let X1, X2, X3, X4, X5 be a random sample from a N(μ,σ2) distribution, where μ and σ are both unknown. Suppose that we have observed the following values 27.72, 22.24, 32.86, 19.66, 35.34. We
Let X1, X2, X3, . . ., Xn be a random sample from this distribution, which means that the Xi's are i.i.d. and Xi ∼ Bernoulli(θ). In other words, we randomly select n voters (with replacement) and
Let X1, X2 ,..., X121 be a random sample from an unknown distribution. After observing this sample, the sample mean and the sample variance are calculated to beDesign a test to decide betweenand
There are two candidates in a presidential election: Candidate A and Candidate B. Let θ be the portion of people who plan to vote for Candidate A. Our goal is to find a confidence interval for θ.
Consider the following observed values of (xi, yi):a. Find the estimated regression linebased on the observed data.b. For each xi, compute the fitted value of yi usingc. Compute the residuals, ei =
Suppose we would like to test the hypothesis that at least 10% of students suffer from allergies. We collect a random sample of 225 students and 21 of them suffer from allergies.a. State the null and
A farmer weigh s 10 randomly chosen watermelons from his farm and he obtains the following values (in lbs): 7.72 9.58 12.38 7.77 11.27 8.80 11.10 7.80 10.17 6.00Assuming that the weight is normally
Consider the simple linear regression modelwhere ϵi's are independent N(0,σ2) random variables. Therefore, Yi is a normal random variable with mean β0 +β1xi and variance σ2. Moreover, Yi's are
Consider the following observed values of (xi, yi): (1, 3), (3, 7).a. Find the estimated regression linebased on the observed data.b. For each xi, compute the fitted value of yi usingc. Compute the
For the data given in Example 8.20, find a 95% confidence interval for σ2. Again, assume that the weight is normally distributed with mean μ and and variance σ, where μ and σ are unknown.Example
You have a coin and you would like to check whether it is fair or not. More specifically, let θ be the probability of heads, θ = P(H). You have two hypotheses: . 0 = 0o = 1/². Ho (the null
The average adult male height in a certain country is 170 cm. We suspect that the men in a certain city in that country might have a different average height due to some environmental factors. We
Let X1, X2,...,Xn be a random sample from a N(μ,σ2) distribution, where μ is unknown and σ is known. Design a level α test to choose between Ho: Ηο: μ Σμο, Η: μ > μο.
You have a coin and you would like to check whether it is fair or biased. More specifically, let θ be the probability of heads, θ = P(H). Suppose that you need to choose between the following
Here, we look again at the radar problem (Example 8.23). More specifically, we observe the random variable X: X = θ + W, where W ∼ N(0,σ2 = 1/9). We need to decide betweenLet X = x. Design a
Let X1, X2,...,Xn be a random sample from a N(μ, σ2) distribution, where μ and σ are unknown. Design a level α test to choose betweenH0: μ = μ0,H1: μ ≠ μ0.
For the above example (Example 8.24), find β, the probability of type II error, as a function of μ.Example 8.24 Let X1, X2,...,Xn be a random sample from a N(μ,σ2) distribution, where μ is
Let X1,X2,...,Xn be a random sample from a N(μ,σ2) distribution, where μ is unknown but σ is known. Design a level α test to choose betweenH0: μ = μ0,H1: μ ≠ μ0.
Again consider the simple linear regression modelwhere ϵi's are independent N(0,σ2) random variables, and Y = Bo + Britai
Consider a radar system that uses radio waves to detect aircraft. The system receives a signal and, based on the received signal, it needs to decide whether an aircraft is present or not. Let X be
Consider the following observed values of (xi, yi): (1, 3) (2, 4) (3, 8) (4, 9)1. Find the estimated regression linebased on the observed data.2. For each xi, compute the fitted value of yi using3.
For the data in Example 8.31, find the coefficient of determination.Example 8.31Consider the fo llowing observed values of (xi, yi):(1, 3) (2, 4) (3, 8) (4, 9)

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