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Statistica - esercizi

Esercizi di Statistica per l’esame della professoressa De Iaco. Gli argomenti trattati sono i seguenti: a help to make inference about a population variance, the probability theory with a fidelity, Hypothesis Testing, CASE 2 (Right Tailed Test), CASE 3 (Two Tailed Test).

Esame di Statistica docente Prof. S. De Iaco

Anteprima

ESTRATTO DOCUMENTO

2

∑ x x

́

( − )

2 i

s = n−1 2

(n−1)s

A help to make inference about a population variance is to use the sampling distribution of :

2

σ

this distribution has a chi-square distribution whenever a simple random sample of size n is selected from a

normal population, so the notation is:

2

s

(n−1) 2

χ distribution

2

σ

We can use the chi-square distribution to develop interval estimates and conduct hypothesis tests about a

population variance.

Interval Estimation

We cannot estimate the exact value of the variance, so we need to compute an interval estimation. Intervals

are commonly chosen such that the parameter falls within with a 95 or 99 percent probability, called

confidence intervals

confidence coefficient. Hence, the intervals are called ; the end points of such an

interval are called upper and lower confidence limits and containing a population parameter that is

established by calculating statistics from values measured on a random sample taken from the population

probability theory

and by applying the with a fidelity that represent those of the entire population. The

probability tells what percentage of the time the assignment of the interval will be correct but not what the

chances are that it is true for any given sample. Of the intervals computed from many samples, a certain

percentage will contain the true value of the parameter being sought.

So we can define the interval estimation as:

− −

2 2

( n 1

) s ( n 1

) s

σ

≤ ≤

2

χ χ

2 2

α α

/ 2 (

1 / 2 )

Proof 2

n−1 s

( )

2 2

≤ ≤❑

1−α α

/2 /2

2

σ

Working on a leftmost quantity: 2

2

n−1 s

( )

2 ≤

1−α /2 2

σ

2 2 2

σ ≤ n−1 s

( )

= 1−α 2

/ 2

n−1 s

( )

2

σ ≤

= 2

1−α /2

For the rightmost quantity the procedure is similar, so we obtain:

2

n−1 s

( ) 2

≤σ

2

α /2

The result of the combination of two expression is:

− −

2 2

( n 1

) s ( n 1

) s

σ

≤ ≤

2

χ χ

2 2

α α

/ 2 (

1 / 2 )

Hypothesis Testing

Hypothesis testing is a method for testing a claim or hypothesis about a parameter in a population, using

data measured in a sample. In this method, we test some hypothesis by determining the likelihood that a

sample statistic could have been selected, if the hypothesis regarding the population parameter were true.

We can use three different types of hypothesis testing:

CASE 1 (Left Tailed Test)

σ σ

2 2

H :

0 0 Hypotheses

σ σ

>

2 2

H :

a 0

− 2

( n 1

) s

χ =

2 σ 2

0 Test Statistic 3

χ χ

>

2 2

α Rejection Rule 2

Reject H if (where is based on a chi-square distribution with n - 1 degrees of freedom)

α

0

or α

Reject H if p-value <

0

CASE 2 (Right Tailed Test)

σ σ

2 2

H :

0 0 Hypotheses

σ σ

<

2 2

H :

a 0

− 2

( n 1

) s

χ =

2 σ 2

0 Test Statistic

χ χ

<

2 2 α

(

1 ) Rejection Rule 2

2 2 ❑

❑ >❑

Reject H if or (where is based on a chi-square distribution with n

α (1−α )

/2

0 α

- 1 degrees of freedom) or Reject H if p-value <

0

CASE 3 (Two Tailed Test)

σ σ

=

2 2

H :

0 0

Hypotheses

σ σ

2 2

H :

a 0

− 2

( n 1

) s

χ =

2 σ 2

0 Test Statistic 4

χ χ

<

2 2 α

(

1 / 2 ) Rejection Rule 2 2

2 2 ❑ ∧❑

❑ >❑

Reject H if or (where are based on a chi-square

2 α

α (1−α )/ /2

/2

0 α

distribution with n - 1 degrees of freedom) or Reject H if p-value <

0

So, we can define now the hypothesis tests about a population variance as follows:

2

n−1 s

( )

2

❑ = 2

σ 0

2

χ

Where has a chi-square distribution with n-1 degrees of freedom

As we’ve seen previously,two tools can be used to check the hypothesis testing:

p-value: p-value estimate the strength of rejection of null hypothesis (H0) when that hypothesis is

 true. The lower p-value, greater the strength of rejection;

critical value approach: involves to determine "likely" or "unlikely" by determining whether or not the

 observed test statistic is more extreme than would be expected if the null hypothesis were true. If the

test statistic is more extreme than the critical value, then the null hypothesis is rejected in favor of the

alternative hypothesis. If the test statistic is not as extreme as the critical value, then the null

hypothesis is not rejected.

Inferences About Two Population Variances

Sometimes, we need to compare the variances concerning to different production process, methods, etc. so

we will be using the random sample data which are independent (one from population 1 and the other from

2 2

s s

population 2). The value of and will be necessary to make inference about the variance of two

1 2

2 2

σ σ

populations, indicated with and respectivly. Every time the two variances are equal, the

1 2

2 2

s s

sampling distribution of the ratio / is defined as follow:

1 2

2

s 1 2 2

n n σ =σ

1 2

with size equal to for the population 1 and for the population 2 and 1 2

2

s 2

In this case:

2 n

s

• 1

is the sample variance for the random sample of from population 1

1

2 n

s

• 2

is the sample variance for the random sample of from population 2

2

n n

−1 −1

With degrees of freedom for the numerator and degrees of freedom for the denominator.

1 2

5


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DETTAGLI
Esame: Statistica
Corso di laurea: Corso di laurea in economia e finanza
SSD:
A.A.: 2014-2015

I contenuti di questa pagina costituiscono rielaborazioni personali del Publisher Stig_ di informazioni apprese con la frequenza delle lezioni di Statistica e studio autonomo di eventuali libri di riferimento in preparazione dell'esame finale o della tesi. Non devono intendersi come materiale ufficiale dell'università Salento - Unisalento o del prof De Iaco Sandra.

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