Random (Casual) Sample and Statistic

Statistical Inference

Random Sample, Parameters, and Statistics

A sample of size \(n\), drawn from a population according to a random (or casual) process, is called a random sample if the values of the random variables \(X_1, X_2, \ldots, X_n\) are independent and identically distributed. The term \(X_1\) denotes the first element of the sample, \(X_2\) the second, and \(X_n\) the \(n\)-th.

Statistics and Estimates

Once a particular sample \(x_1, x_2, \ldots, x_n\) has been observed, it is possible to obtain the value of the statistic associated with this sample, namely \(T(x_1, x_2, \ldots, x_n)\), yielding an estimate for the parameter.

Exercise I

Consider \(X_1, X_2, \ldots, X_n\), \(n \in \mathbb{N}\), a random sample from a \(Normal(\mu = -2, \sigma)\) population, where \(\sigma\) is unknown. The following r.v.’s are not statistics:

1. [T] [F] F \(\quad T(X_1, X_2, \ldots, X_n) = X_1 + X_2 + \cdots + X_n\)

2. [T] [F] F \(\quad T(X_1, X_2, \ldots, X_n) = \bar{X}\)

3. [T] [F] T \(\quad T(X_1, X_2, \ldots, X_n) = \dfrac{\bar{X}}{\sigma}\)

4. [T] [F] F \(\quad T(X_1, X_2, \ldots, X_n) = \displaystyle\frac{\sum \left(X_i - \mu\right)^2}{n}\)

5. [T] [F] F \(\quad T(X_1, X_2, \ldots, X_n) = \dfrac{\sum (X_i - \bar{X})^2}{n-1}\)

6. [T] [F] T \(\quad T(X_1, X_2, \ldots, X_n) = \displaystyle\sum \left(\dfrac{X_i - \mu}{\sigma}\right)^2\)

Exercise II

Let \(X_1, X_2, \ldots, X_n\), \(n \in \mathbb{N}\), be a random sample from a population \(X\) (with unknown probabilistic behavior), with expected value \(\mu\) and variance \(\sigma^2\), both known.

1. The expected value of the statistics \(\displaystyle\sum_{i=1}^{n} X_i\) and \(\bar{X}\) are, respectively, equal to:
   a)\(n\mu\) and \(\mu\) \(\quad\) b) \(n\mu\) and \(\mu/n\) \(\quad\) c) \(\mu\) and \(\mu/n\) \(\quad\) d) none of the above

Chi-Square Distribution

A pertinent question arises: How to obtain the sampling distribution of a statistic or of a random variable?

Chi-Square Distribution: Main Properties (1)

Chi-Square Distribution: Main Properties (2)

1. The graphical shape of the distribution depends on its degrees of freedom: \(n\) (\(n \in \mathbb{N}\)).

2. It is a positive and non-symmetric function. The skewness decreases as the degrees of freedom increase.

3. If \(X \sim \chi^2_{(n)}\), then: \(E(X) = n\) and \(Var(X) = 2n\).

4. It is an additive distribution. The sum of \(k\) independent random variables, where \(X_i \sim \chi^2_{(n_i)}\), is still a random variable such that: \[\sum_{i=1}^k X_i \sim \chi^2_{\left(n = \sum_{i=1}^k n_i\right)}\]

Exercise V

Let \((Z_1, Z_2, Z_3, Z_4, Z_5)\) be a random sample from a \(N(0, 1)\) population and \(Q = \sum_{i=1}^5 Z_i^2\).

1. [V][F] \(\quad P(1.610 < Q < 9.236) = 0.8\)

2. [V][F] \(\quad\) The standard deviation of \(Z_3^2 + Z_4^2\) is equal to 4.

Student’s t Distribution

Let \(X \sim Normal(0, 1)\) and \(Y \sim \chi^2_{(n)}\) be independent random variables. Then the random variable:

\[T = \frac{X}{\sqrt{Y/n}} \sim t_{(n)}\]

has a Student’s \(t\) distribution with \(n\) degrees of freedom, i.e.: \(T \sim t_{(n)}\).

Student’s t Distribution: Properties

2. It is a symmetric function around the mean.

3. If \(X \sim t_{(n)}\), then: \(E(X) = 0\) and \(Var(X) = \dfrac{n}{n-2}\), with \(n > 2\).

Student’s t Distribution: CLT Approximation

4. With increasing degrees of freedom, the Student’s \(t\) distribution tends to the Normal distribution, by the C.L.T.:
   \[X \;\overset{\text{CLT}}{\dot\sim}\; N\left(0, \;\frac{n}{n-2}\right) \iff Z_n = \frac{X}{\sqrt{\frac{n}{n-2}}} \;\overset{\text{CLT}}{\dot\sim}\; N(0, 1)\]

Case 1: Normal Population, \(\sigma^2\) Known

Population: \(X \sim N(\mu, \sigma)\), with \(\sigma^2\) known.

From a random sample \(X_1, X_2, \ldots, X_n\), where \(X_i \sim N(\mu, \sigma)\), we obtain the statistic sample mean:

\[\bar{X} = \frac{1}{n}\sum_{i=1}^n X_i\]

Case 2: Normal Population, \(\sigma^2\) Unknown

Population: \(X \sim N(\mu, \sigma)\), with \(\sigma^2\) unknown.

Statistic: Mean of a random sample of size \(n\): \(\bar{X} = \dfrac{1}{n}\displaystyle\sum_{i=1}^n X_i\)

Case 2 (cont.) and Case 3

\[S^2 = \frac{1}{n}\sum_{i=1}^n (X_i - \bar{X})^2 \qquad \text{and} \qquad S'^2 = \frac{1}{n-1}\sum_{i=1}^n (X_i - \bar{X})^2 = \frac{n}{n-1}S^2\]

Case 3 (cont.)

Approximate sampling distribution of \(\bar{X}\): by the C.L.T.,

\[Z = \frac{\bar{X} - \mu}{\dfrac{\sigma}{\sqrt{n}}} \;\overset{\text{CLT}}{\dot\sim}\; N(0, 1)\]

Exercise VII

A certain company intends to validate the value (in euros) of its accounts receivable from clients, with an underlying distributional behavior of the account values being \(Normal(\mu = 385, \sigma = ?)\). A random sample of 25 client accounts (independent) was selected, yielding a corrected sample standard deviation of \(s' = 122.6\) euros.

Exercise VII — Solution: a)

Population: \(X \sim Normal(\mu = 385, \sigma = ?)\) (standard deviation unknown)

Sample: \(n = 25 \rightarrow s' = 122.6\)

Exercise VII — Solution: b)

b)

Population: \(X \sim Normal(\mu = 385, \sigma = ?)\) (standard deviation unknown)

Sample: \(n \geq 30 \rightarrow s' = 122.6\)

Sampling Distribution of \(S^2\)

Population: \(X \sim N(\mu, \sigma)\), with mean \(\mu\) unknown and variance \(\sigma^2\).

Selected a random sample \(X_1, X_2, \ldots, X_n\), where \(X_i \sim N(\mu, \sigma)\), we obtain the statistic sample variance \(S^2\) or corrected sample variance \(S'^2\):

Exercise VIII

A pharmaceutical company produces pills in which the variability of the quantity of the active substance from one pill to another should be very small. The standard deviation of the population is, supposedly, one milligram. Inspectors from the Ministry of Health selected a random sample of 16 pills. Assuming the population is Normal, the probability that the corrected sample variance is greater than \(0.736\;mg^2\) is equal to:

1. 0.76 \(\qquad\) b) 0.25 \(\qquad\) c) 0.75 \(\qquad\) d) none of the above

Case 1: Total of Successes

Selected a random sample \(X_1, X_2, \ldots, X_n\) from a Bernoulli population, where \(X_i \sim Bernoulli(1, p)\), and let \(Y\) be the number of elements that possess the characteristic of interest (successes) in the sample of size \(n\).

Case 2: Sample Proportion

Case 2:

Let the statistic \(\bar{X} = \dfrac{1}{n}\displaystyle\sum_{i=1}^n X_i = \dfrac{Y}{n}\), the proportion (relative frequency) of successes that, in a sample of size \(n\), possess a given attribute.

Case 2: Approximate Distribution of \(\hat{p}\)

When \(n \to \infty\), the C.L.T. yields the approximate sampling distribution:

\[\bar{X} = \hat{p} \;\overset{\text{CLT}}{\dot\sim}\; N\left(p, \;\sqrt{\frac{p(1-p)}{n}}\right) \iff Z_n = \frac{\bar{X} - p}{\sqrt{\dfrac{p(1-p)}{n}}} \;\overset{\text{CLT}}{\dot\sim}\; N(0, 1)\]

Exercise X (Statement)

In a certain neighborhood, the proportion of inhabitants who consider the opening of a new supermarket useful is 0.8. The supermarket chain Gota Doce intends to open a new store in this neighborhood. Nevertheless, before making a decision, a study was conducted in which 400 inhabitants were randomly interviewed to assess their intention to use the services of the new supermarket.

Exercise X — Solution: a)

\(p = 0.8\) (proportion of inhabitants who consider the opening of a new supermarket useful)

a) \(X \to\) number of inhabitants, out of 400, who consider the opening of the new supermarket useful.

\[X \sim Binomial(n = 400, p = 0.8)\]

Exercise X — Solution: b)

b) Of the interviews conducted, \(X\) inhabitants out of \(n\) expressed the opening of Gota Doce was useful, so the sample proportion is \(\frac{X}{n}\).