Sampling Distributions

IT233: Applied Statistics TIHE 2005

Random Sample:

A random sample is one in which each unit of the population has an equal chance of being selected in the sample.

x, x ,..., x A set of observations ( 1 2 n ) constitutes a random sample if the following are true.

1. Each observation has been selected at random from the same population.

2. The observations are independent (i.e. any selection is not influenced by the previous selections).

x, x ,..., x NOTES: To ensure that 1 2 n are indeed independent and identically distributed we follow the following notes:

1. To obtain a random sample we should be sampling from an infinite population.

2. If we are sampling from a finite population, the selection should be done with replacement.

Sample Statistic:

Any quantity that computed from a random sample is known as statistic.

Examples: Common statistics are: x 1. The sample mean: x = i n 2 ( x- x ) 2. The sample variance: S2 = i n -1

3. The sample median: x

Q, Q 4. Sample quartiles: 1 3

5. Sample range: Largest – Smallest

x - m m NOTE: The r. v. ( i ) is not a statistic. Since is unknown, we cannot determine its numerical values from a random sample.

Sampling Distribution:

Since a sample statistic is computed from a random sample, it follows that the sample statistic itself is random variable. Every random variable has a probability distribution. This distribution of a sample statistic is called its sampling distribution.

Definition: The probability distribution of a sample statistic is called a sampling distribution.

Some Sampling Distributions

 Sampling distribution of x x- x  Sampling distribution of the difference of two means ( 1 2 )  Sampling distribution of S2 S 2 1  Sampling distribution of the ratio of two sample variances – 2 S2 Sampling Distribution of x :

If a random sample of n observations is taken from a normal population with mean m and variance s 2 , then

m= m x

s 2 s 2 = x n

and its standard normal value z follows standard normal distribution i.e.

x - m z= : N(0,1) s n

NOTE: If the sample is drawn from unknown distribution, either finite or infinite, the distribution of x will still be approximately normal provided n is large (n >30), which is given by the following theorem:

Central Limit Theorem: If x is the mean of a random sample of size n taken from a population with mean m and variance s 2 , then

x - m z= : N(0,1) s n as n  .

Applications: The sampling distribution of x could be used:

 To find the probabilities concerning x  To find the confidence interval for m H : m= m  To test the hypothesis 0 0

Example: A random variable X has the following distribution: x 4 5 6 7 p( x )= P( X= x ) 0.2 0.4 0.3 0.1 (a) Find m and s 2 of X . m 2 (b) Find the mean x and variance s x for sample size 36. (c) Find the probability that sample mean x will be less than 5.5.

Solution:

(a) Mean m

m =E( X ) = x p i i i = 5.3

Variance s 2

2 2 E( X ) = xi p i = 28.9 i

\ 2 2 2 s =E( X ) - 臌轾 E ( X ) = 28.9 – 28.09 = 0.81

m= m (b) x = 5.3

s 2 0.81 s 2 = = = 0.0225 x n 36

(c) The distribution of X is not normal but n >30. Hence, we can take the distribution of X to be normal.

骣 5.5- 5.3 = 0.9082 \ P( X< 5.5) = P琪 Z < 桫 0.0225

= P( Z < 1.33) 0 1.33 z