Statistics and Probability Answers

A firm purchases a microchip used in the manufacturing of LED screens from four different
suppliers. TechZone supplies 30 percent of the total microchips, Advance Electronics 25 percent,
PSP importers 18 percent, and TechParts inc. 32 percent. TechZone tends to have the best
quality, as only 3 percent of its microchips arrive defective. Advance Electronics chips are 4
percent defective, PSP importers chips are 7 percent defective, and Parts Inc. are 6.5 percent
defective.
a) If a random chip is selected, what is the probability that the chip is defective
b) A defective chip was discovered in the latest shipment. What is the probability that it came
from TechZone supplies?
c) A defective chip was discovered in the latest shipment. What is the probability that it came
from Parts Inc.?

Suppose you start with one penny and repeatedly flip a fair coin. Each time you get heads, before the first time you get tails, you get two more pennies. Let X be the total number of pennies you have at the end. Compute E(X).

Suppose that Y1, Y2,...,Yn form a random sample from a PARETO (α, β) distribution, where α and β are unknown. Show that the maximum Likelihood Estimates of α and β (i.e. α-hat and β-hat) can be found by solving the following set of equations:

(n/α) + n ln(β-hat) = Σni=1 ln[β-hat + Yi]
(nα/β-hat) = (α-hat+1) Σni=1 [β-hat + Yi]^-1

NOTE: The pdf of a PARETO (α, β) distribution is given by

f(y) = {(αβα)/(β+y)^(α+1) if y>0}
{0 if y≤0}

Four of the Statistics for Business and Economics students submitted an entry to a Math contest at the faculty. A total of 24 entries were submitted. Six of the entries will move on to the next round.

a. What is the hypergeometric probability that all four Statistics for Business and Economics students will move on to the next round? (2 Marks)
b. How many Statistics for Business and Economics students are expected to move to the next round?
(2 Marks)
c. What is the hypergeometric probability that fewer Statistics for Business and Economics students than expected will make it to the next round?

In a local referendum to legalize the sale of beer, voters were surveyed about their opinions on the referendum and their views on the moral issue of selling alcohol beverages. From the contingency table, test at 0.05 alpha whether their moral values are related to their opinion on the referendum:



Opinion on referendum



Drinking is :



For Against Undecided

OK

95

83

21

Tolerated

73

71

18

Immoral

12

46

8



Formulate the H0 and H1
C.V. at alpha 0.05 level of significance
Compute the test value: Chi-square X2
Decision
Interpretation

Super bikes arrive in a 12 lot showroom. Inspections are carried out on six out of every 12. For
one lot, it is known 2 out of 12 do not meet the safety standards prescribed. What is the
probability that at least 2 out of the 6 that were tested from that lot will not meet safety
standards?

A firm wants to estimate mean lifetime of a tool.From previous experiments it is known that lifetimes are normally distributed.It draws a random sample of four of these tools and finds that there lifetimes are 9.3,7.9,10.8and 11.4.Calculate a 95% cinfidence interval for mean lifetime of that kind of tool

Statistics department data show that 20% of Pakistani age 18 and over holds a certificate of
matriculation. If 10 Pakistani national age 18 and above are randomly selected, find the
probability that at least 3 will have certificates of matriculation.

The following is a breakdown of faculty members at IIM-A.
Rank No of males No of females
Professor 20 12
Associate Prof 18 20
Asst Prof 25 30.
What is the probability that a faculty member selected at random is
a) Female.
b) A female professor
c) A female or professor.
d)A professor or asst. Prof

The height of the students in a certain class is following a normal distribution with mean height as 165 cm and a standard deviation of 25 cm. There are 60 students in that class. Determine

i. The number of students whose height is more than 158 cm. (5 Marks)

ii. The number of students whose height is lying between 155 and 172 cm. (5 Marks)

The answer should be a min of 800 words

3. b. Find the lowest height among the tallest 5 students in a class.