Chapter I - Statistics II - Sampling Distributions

This is a set of exercises for Statistics II, Chapter II — Sampling Distributions, from the Lisbon Accounting and Business School (ISCAL–IPL).

Contents: t-Student and Chi-Squared distributions. Random sample and statistic. Sampling distributions: of the mean in Normal and non-Normal populations, of the variance in Normal populations, and of the proportion from a Bernoulli population.

The exercise set consists of True/False questions (where you should determine the truth value and justify your choice) and multiple choice questions (where you should select the correct option and justify it appropriately).

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Questions

Question 1

Let \((X_1,X_2,\ldots,X_n)\) with \(n\in\mathbb{N}\), a random sample from a population distributed \(X\sim Normal(\mu=-2,\sigma)\), where \(\sigma\) is unknown. Asses if, the following random variables are not statistics, i.e., if you select true, you are claiming that the random variable is not a statistic.

1. TRUEFALSE \(T(X_1,X_2,\ldots,X_n)=X_1+X_2+\ldots+X_n\)

2. TRUEFALSE \(T(X_1,X_2,\ldots,X_n)= \frac{X_1+X_2+\ldots+X_n}{n}=\bar{X}\)

3. TRUEFALSE \(T(X_1,X_2,\ldots,X_n)= \frac{\bar{X}}{\sigma}\)

4. TRUEFALSE \(T(X_1,X_2,\ldots,X_n)= \frac{\bar{X}-\mu}{\sigma}\sqrt{n}\)

5. TRUEFALSE \(T(X_1,X_2,\ldots,X_n)= \frac{\sum_{i=1}^n(X_i-\mu)^2}{n}\)

6. TRUEFALSE \(T(X_1,X_2,\ldots,X_n)= \frac{\sum_{i=1}^n(X_i-\bar{X})^2}{n}=S^2\)

7. TRUEFALSE \(T(X_1,X_2,\ldots,X_n)= \frac{\sum_{i=1}^n(X_i-\bar{X})^2}{n-1}=S'^2\)

Question 2

Let the waiting time for take-off of each plane in a given airport is a r.v. distributed \(N(\mu=4,\sigma)\). Randomly chosen 21 flights, the records allow to estimate \(s'=1.81min\).

TRUEFALSE The probability that the average waiting time, from the sample, is larger than 5 minutes equals \(0.010\).

Question 3

In a given financial firm, the profits obtained from each investment, in the last year, average 8,200 euro, and a standard deviation of 3,600 euro. Consider a random sample of 100 investments. The probability that the sample average is farther away from the true mean by 806.76 euro, is: 0.01250.0250.9875None of the previous

Question 4

A given company intends to check the value (in euro) of the receivable accounts, with an underlying statistical behavior that follows a Normal distribution with mean \(\mu=385\) and standard deviation \(\sigma\). Choosing randomly a sample with \(n\) accounts (independently), you obtained a (corrected) sample standard deviation of \(122.6\) euro.

TRUEFALSE The size of the sample to collect (assuming that the estimate for \(s'\) is maintained) such that the sample mean is not farther from the true mean than 20 eur in 90% of the cases is \(n=102\). (Assume for your computations, that the value of \(n\) will be larger than 30)

Question 5

Assume that a particular streaming service is subscribed by 10% of the population in a given country. Because this service is very recent, a marketing firm decided to estimate that proportion, based in a sample of 100 citizens, before renewing their contract with the streaming service company.

TRUEFALSE Knowing that the contracts will be renewed only if the sample proportion is larger than 8.5%, the probability tha this contract will be renewed is 0.6915.

Question 6

Assume that you intend to study the feasibility of a project which core variable is the price (in monetary units) of raw-materials, which is well represented by the random variable \(X\sim N(\mu,\sigma)\). It was selected, randomly, a sample \((X_1,\ldots,X_{16})\) from that population, where the sample (corrected) variance was found to be \(0.18 um^2\).

1. TRUEFALSE \(Q=\frac{15S'^2}{\sigma^2}\sim \chi^2_{16}\)
2. TRUEFALSE \(P(S'2^2>1.2163 \sigma^2)=0.25\)

Question 7

15 adult males, with ages between 35 and 50, participated in a study to assess the effect of diet and exercises in the cholesterol levels found in blood. The cholesterol was measured at the begining and after three months after participating of a program that consisted in a new diet and aerobic exercises. The data is found in the following table:

Individual	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
Before	265	240	258	295	251	245	287	314	260	279	283	240	238	225	247
After	229	231	227	240	238	241	234	256	247	239	246	218	219	226	233

Assume that:

• The samples Before and After, are paired (each column is the same individual)
• Each sampled pair, is an independent observation of the other.
• The cholesterol measures follow and Normal distribution.

TRUEFALSE The probability that the mean of the difference between after and before the treatment is farther away from its mean by more than 6.61 is 0.2.