Statistics and Probability Answers

the first moment of a distribution about the value 1 is 4 then mean is

company produces low-energy light bulbs. The bulbs are described as using 9 watts of power. Amir, the production manager, asked Jenny to measure the power used by each of a sample of 120 bulbs from the latest batch produced and to test the hypothesis that the mean for the batch is 9.0 watts. Jenny is to carry out the test at the 5% significance level. (a) What assumption must be made about the sample of 120 bulbs if the result of this test is to be valid? (b) Jenny found that the mean for the sample was 9.2 watts and that the standard deviation was 1.3 watts. Carry out the test asked for by Amir. (c) When Jenny reported the conclusion of the test to Amir, he said that he had intended to ask Jenny to test whether the bulbs in the batch use more than 9.0 watts on average. For the test in part (b), state the effect that this new information would have on: (i) the alternative hypothesis; (ii) the critical value(s); (iii) the conclusion.

Evaluate probability theory to an example involving hashing and load balancing.

It is known that the average hemoglobin of normal adult men is 140g/L. This research randomly investigates 60 adult men in a factory. The mean hemoglobin is 125g/L and the standard deviation is 15g/L. A researcher considered that the average hemoglobin of adult men in this factory is lower than that of general adult men. (1) Is this conclusion correct? Why? (10 points) (2) If we want to estimate the average hemoglobin of adult men in this factory, what statistical analysis method can we use? What are the results? (20 points) (3) To compare the average hemoglobin of adult men in this factory with that of normal adult men, what method can we use? Write down the complete analysis steps and draw a conclusion. ,

Random variable X has the following probability distribution
X=X -2 -1 0 1
2 3
PAX) 0.1 0.2 2 03 3k
Find
the value of k
(ii) Evaluate P(X2)&P(-2<X2)
Find The Cumulative Distribution Of X
(iv)
Evaluate the Mean of X

Let Xk (k = 1, . . . , n) be n independent random variables such that
(a) If we interpret Xk to be the number of heads on the kth toss of a coin, what interpretation can be given to Sn= X1+...,+Xn?

(b) Show that the law of large numbers in this case reduces to lim n to infinity P (|Sn/n-p|>/epsolom)=0
and interpret this result

A casino offers the following game: you draw one card from a standard 52-card deck. If you draw a jack, you win $1.25. If you draw a queen, you win $3.25. If you draw a king, you win $5.25 dollars. If you draw any ace except the ace of spades, you win $6.75. If you draw the ace of spades, you win $8.5.

The entry fee to play this game is $2. Compute the expected value of this gamble (include the entry fee in your expected value).

The following sets are used in a company's database:
Workforce = {Abi, Baz, Cal, Dev, Eva, Fay, Ged}
Department = { Accounts, Operations, Research, Sales}
Place = {Sheffield, Leeds, London}
ProjectTeams = {A, B, C}
A = {Abi, Eva, Fay}
B = {Baz, Dev, Eva, Fay}
C = {Cal, Eva, Ged}
Find the following:
(a) A ∩ B
(b) B ∪ C
(c) A ∩ B ∩ C
(d) Workforce \ B
(e) Workforce \ (A ∪ C)
(f) (Workforce \ B) \ C
(g) The power set P(A)
(h) The Cartesian product Place x ProjectTeams
(i) The number of elements in the power set P(Department)
(j) P(A) ∪ A

Q-3 The daily COVID 19 cases (in hundred) for Delhi for past 2 week is summarizing in the following table:
Day 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Cases 28 29 33 31 37 34 36 43 41 32 34 37 39 32

a. Using exponential smoothing method forecast the cases for 15 days, taking alpha as 0.3 and Initial forecast is the average of all data.
b. Using linear trend analysis, find the trend line for number of COVID 19 cases in Delhi and forecast for next 3 days. Also compute the Mean Square Error.