Question

Ahmed àon the paper providenk, do not use your own pager. Work only on ne proteem on each aheet the tue ides of the some of the problens you are working on by Awer te Part b)" tthe wamber of the problem top of the page). Do not une a red pen or a red pencil. Do not write in the corner covered t that you up by the same amount of credit Show all your work. the staple (top left corner on the front side, top right corner on the back side Each problem is worth 1. The numbers 811 84.7, 77.8, 76.4 represent the temperature measured at a certain location and at a certain time. This is regarded as a random saanple from a distribution Arm,e*), where m is the true temperature, and σ' measures the precision of the method. a) Find an unbiased estimate for m. 6) Find an unbiased emtimate for q it it is known that m-79.0. 2. The numbers 81.1, 84.7, 77.8 76.4 represent the temperature measured at a certain location and at a certain time. This is regarded as a rando sample from a distribution N (n, 16), where m is the true temperature, and -16 measures the precision of the method: the value of m is not known. Find a 90% confidence interval for m. 3. The numbers 81.1, 84.7, 77.8, 76.4 represent the temperature measured at a certain location and at a certain time. This is regarded as a random sample from a distribution M m, σ2), where m is the true temperature, and σ measures the precision of the method; the values of m and σ2 are not known. Find a 90% confidence interval for m. 4. The numbers 81.1, 84.7, 77.8, 76.4 location and at a certain time. This is regarded as a random represent the temperature measured at a certain sample from a distribution M m,ơ2), where rn is the true temperature, and o? measures the precision of the method: the values of m and ơ2 are not known. a) Find a 90% confidence interval for σ2 b) Find a 90% confidence interval for σ. 5. In a random sample of 500 persons eating lunch at a department store cafeteria, 204 persons had desert. Construct a 90% confidence interval for the true proportion of persons usually eating dessert in this cafeteria.

Answer 1

Q1)

a) sample mean is unbiased estimator for m

= 1/4 ( 81.1 + 84.7 + 77.8 + 76.4) = 80

b)

sample variance is unbiased estimator of $\sigma^2$

sample variance = 13.7

c)

answer is same as b) part

Prove

n. yi 34. μ.

for b) and c)

(n - 1)s2

$E( \chi ^2_{n-1}) = (n-1)$   as E(chi-square with n degree of freedom ) is degree of freedom

hence

o2

o2

DE-1 o2

$E\left ( \frac{ s^2}{\sigma^2} \right )= 1$

$E (s^2) = \sigma ^2$