Probability And Statistics | Week 10

Probability And Statistics Week 10 Answers

Q1. Nine wafers are baked at two different temperatures, 90°C and 100°C, to examine the impact of temperature on the weights (in gm) of wafers of a specified size. Find a 95% confidence interval on the difference in means assuming that the measurements in the two columns follow independent normal distributions with distinct variances.

(A) (-0.25, 3.72)
(B) (-0.28,3.16)
(C) (0.28,3.21)
(D) (0.25,3.85)

Q2. Suppose X₁, X₂,…., Xm are m independent and identically distributed random samples from a uniform [0, θ] distribution. Which of the following is/are unbiased estimator(s) of θ?
(A) 2X
(B) m+1/m X(m)

(C) X
(D) m/m -1 X(m)
Here X = 1/mΣm i=1 Xi and X(m) = max Xi.

Q3. Let X₁,…, Xn be independent and identical random samples from a gamma distribution with pdf
f(x; a) = 1/acr(c) -xc-¹e-(a), x ≥ 0,
where c> 0 is known. Which of the following is UMVUE of a?

(A) X

(B) X/c
(C) Σn i=1 Xi
(D) 1/cΣn i=1 Xi

Q4. A group of 10 patients has mean weight 230 pounds. The sample standard deviation is given as 20 pounds. What is the 95% confidence interval for a sample for the true mean weight?
(A) (218.407,241.593)
(B) (217.604,242.396)
(C) (215.694,244.306)
(D) (219.565,240.436)

Q5. For a normal population with known standard deviation o, what is the confidence level for the interval
(A) 0.1974
(B) 0.5987
(C) 0.6046
(D) 0.4013

Q6. Independent random samples of sizes n₁ and n₂ are taken from two normal populations with variances of σ²1 and σ²2 respectively. Let n₁ = 15 and n₂ = 10 and the sample variances from the two samples be 25.7 and 29.2 respectively. Find a 90% confidence interval of σ²2/σ²2.
(A) (0.4294,3.4375).
(B) (0.4924,3.4735).
(C) (0.3294,2.3375).
(D) (0.2154,4.9275).

Q7. Let X₁,…, Xm be a random sample from Bernoulli distribution with parameters p, 0 < p < 1. Which of the following is an unbiased estimator of p(1-p)?
(A) m/m-1 X(1 + X)
(B) m/m-1 X(1 – X)
(C) m/m+1 X(1 + X)
(D) m/m+1 X(1 – X)

Q8. Consider eight pairs of zinc concentration of in bottom water and surface water are as follows

Construct 95% confidence interval for mean difference between bottom and surface water.
(A) (0.075,0.222)

(B) (0.055,0.239)
(C) (0.315,0.689)
(D) (0.072,0.568)

Q9.

Q10. A farmer weighs 11 randomly chosen pumpkins from his farm and obtained the weights (in lbs) are
7.21 8.52 6.92 11.15 14.35 7.58 6.21 9.67 10.12 12.42 13.35
Find 95% confidence interval for population variance (σ²) assuming the the weight is normally distributed with unknown mean μ and unknown variance σ².

(A) (5.84,21.92)
(B) (3.74,23.59)
(C) (4.82,21.64)
(D) (2.31,17.56)

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