Probability And Statistics | Week 10
Probability And Statistics Week 10 Answers
Link : Probability And Statistics (nptel.ac.in)
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)
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?
(B) m+1/m X(m)
(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?
(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?
Q5. For a normal population with known standard deviation o, what is the confidence level for the interval
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.
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.
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.