Statistics and Probability Answers

A3. Let (Ω, F, P) be a probability space and let H ∈ F with P(H) > 0. For any arbitrary

A ∈ F, let

PH(A) =

P(A ∩ H)

P(H)

Show that (Ω.F, PH) is a probability space.

In an experiment of tossing a fair coin four times. Let the sample space Ω be the

number of tails observed and ϕ be the impossible event.

(a) List the Ω and Sigma field F, with the maximum cardinality.

(b) If A1, A2, A3, A4 are subsets of Ω, show that the class of sets F = {ϕ, A1, A2, A3, A4, Ω}

is a σ − f ield.

(c) If P is a function defined on F, what properties must P satisfy for the triple

(Ω, F, P) to be called a probability space.

How many ways can 4 baseball players and 3 basketball players be selected from 12 baseball players and 9 basketball players? 

A study by Thienprasiddhi et al. (A-4) examined a sample of 16 subjects with open-angle glaucoma

and unilateral hemifield defects. The ages (years) of the subjects were:

62 62 68 48 51 60 51 57

57 41 62 50 53 34 62 61

Can we conclude that the mean age of the population from which the sample may be presumed to have been drawn is less than 60 years? Let alpha = 0.5

Independent random samples from normal distributions with equal variance of the amount of insider

trading during mergers by employees of investment banking firms (1) and brokerage houses (2) gave the

results listed in the accompanying table.

Investment Banking Firm 10 5 5 5 8 8 8

Brokerage House 6 0 5 2 2 3 3

(a) Find the 95% confidence interval estimate for the difference between means µ1 − µ2..

(b) Would you conclude that employees of Investment banking firms cause more insider trading than

their counterparts in Brokerage House?

Conduct Hypothesis Testing Using Small-Sample Tests. Step- By- Step.

An experimental study was conducted by a researcher to determine if new time slot has an effect on the performance of students in mathematics. Ten randomly selected students participated in the study. Toward the end of the investigation, a standardized assessment was conducted. The sample mean was X=75 and the standard deviation s=20. In the standardization of the test, the mean was 65 and the standard deviation was 10. Based on the evidence at hand, is the new time slot effective? Use α=0.05.

A survey of 100 similar-sized hospitals revealed a mean daily census in

the pediatrics service of 27

with a standard deviation of 6.5. Do these data provide sufficient evidence to

indicate that the

1. population mean is greater than 25? Let alpha = 0.5

We wish to know if we can conclude that the mean daily caloric intake in

the adult rural population of

a developing country is less than 2000. A sample of 500 had a mean of 1985

and a standard deviation

of 210. Let alpha = 0.5

A sample of 25 freshman nursing students made a mean score of 77 on a

test designed to measure

attitude toward the dying patient. The sample standard deviation was 10. Do

these data provide

sufficient evidence to indicate, at the .05 level of significance, that the

population mean is less than

80? What assumptions are necessary?

Nine laboratory animals were infected with a certain bacterium and then

immunosuppressed. The

mean number of organisms later recovered from tissue specimens was 6.5

(coded data) with a

standard deviation of .6. Can one conclude from these data that the

population mean is greater than 6?

Let alpha = 0.5What assumptions are necessary?