SAMPLING DISTRIBUTION Introduction In real life calculating parameters

SAMPLING DISTRIBUTION

Introduction • In real life calculating parameters of populations is usually impossible because populations are very large. • Rather than investigating the whole population, we take a sample, calculate a statistic related to the parameter of interest, and make an inference. • The sampling distribution of the statistic is the tool that tells us how close is the statistic to the parameter. 2

Sampling Distribution of the Mean • An example – A die is thrown infinitely many times. Let X represent the number of spots showing on any throw. – The probability distribution of X is x p(x) 1 2 1/6 3 4 5 1/6 1/6 E(X) = 1(1/6) + 2(1/6) + 3(1/6)+ 6 …………………. = 3. 5 1/6 V(X) = (1 -3. 5)2(1/6) + (2 -3. 5)2(1/6) + …………. …= 2. 92 3

Throwing a die twice – sample mean • Suppose we want to estimate from the mean of a sample of size n = 2. • What is the distribution of ? 4

Throwing a die twice – sample mean 5

The distribution of when n = 2 E( ) =1. 0(1/36)+ 1. 5(2/36)+…. =3. 5 6/36 5/36 V(X) = (1. 03. 5)2(1/36)+ (1. 5 -3. 5)2(2/36). . . = 1. 46 4/36 3/36 2/36 1 1. 5 2. 0 2. 5 3. 0 3. 5 4. 0 4. 5 5. 0 5. 5 6. 0 6

Sampling Distribution of the Mean 6 7

Sampling Distribution of the Mean Notice that is issmallerthan. x. The larger the sample size the smaller. . Therefore, tends to fall closer to , as the sample size increases. 8

SAMPLING DISTRIBUTION • Let X 1, X 2, …, Xn be a r. s. of size n from a population and let T(x 1, x 2, …, xn) be a real (or vector-valued) function whose domain includes the sample space of (X 1, X 2, …, Xn). Then, the r. v. or a random vector Y=T(X 1, X 2, …, Xn) is called a statistic. The probability distribution of a statistic Y is called the sampling distribution of Y. 9

SAMPLING DISTRIBUTION • The sample mean is the arithmetic average of the values in a r. s. • The sample variance is the statistic defined by • The sample standard deviation is the statistic defined by S. 10

SAMPLING FROM THE NORMAL DISTRIBUTION Properties of the Sample Mean and Sample Variance • Let X 1, X 2, …, Xn be a r. s. of size n from a N( , 2) distribution. Then, 11

SAMPLING FROM THE NORMAL DISTRIBUTION • Let X 1, X 2, …, Xn be a r. s. of size n from a N( , 2) distribution. Then, • Most of the time is unknown, so we use: 12

SAMPLING FROM THE NORMAL DISTRIBUTION In statistical inference, Student’s t distribution is very important. 13

SAMPLING FROM THE NORMAL DISTRIBUTION • Let X 1, X 2, …, Xn be a r. s. of size n from a N( X, X 2) distribution and let Y 1, Y 2, …, Ym be a r. s. of size m from an independent N( Y, Y 2). • If we are interested in comparing the variability of the populations, one quantity of interest would be the ratio 14

SAMPLING FROM THE NORMAL DISTRIBUTION • The F distribution allows us to compare these quantities by giving the distribution of • If X~Fp, q, then 1/X~Fq, p. • If X~tq, then X 2~F 1, q. 15

CENTRAL LIMIT THEOREM If a random sample is drawn from any population, the sampling distribution of the sample mean is approximately normal for a sufficiently large sample size. The larger the sample size, the more closely the sampling distribution of X will resemble a normal distribution. Random Sample (X 1, X 2, X 3, …, Xn) X Random Variable (Population) Distribution Sample Mean Distribution 16

Sampling Distribution of the Sample Mean If X is normal, is normal. If X is non-normal, is approximately normally distributed for sample size greater than or equal to 30. 17

EXAMPLE 1 • The amount of soda pop in each bottle is normally distributed with a mean of 32. 2 ounces and a standard deviation of 0. 3 ounces. – Find the probability that a bottle bought by a customer will contain more than 32 ounces. – Solution • The random variable X is the 0. 7486 amount of soda in a bottle. x = 32. 2 18

EXAMPLE 1 (contd. ) • Find the probability that a carton of four bottles will have a mean of more than 32 ounces of soda per bottle. • Solution – Define the random variable as the mean amount of soda per bottle. 0. 9082 0. 7486 x = 32. 2 19

Sampling Distribution of a Proportion • The parameter of interest for nominal data is the proportion of times a particular outcome (success) occurs. • To estimate the population proportion ‘p’ we use the sample proportion. The number of successes The estimate of p = p^ = X n 20

Sampling Distribution of a Proportion • Since X is binomial, probabilities about p^ can be calculated from the binomial distribution. • Yet, for inference about^p we prefer to use normal approximation to the binomial whenever it approximation is appropriate. 21

Approximate Sampling Distribution of a Sample Proportion • From the laws of expected value and variance, it can be shown that E( ) = p and V( )=p(1 -p)/n • If both np ≥ 5 and n(1 -p) ≥ 5, then • Z is approximately standard normally distributed. 22

EXAMPLE – A state representative received 52% of the votes in the last election. – One year later the representative wanted to study his popularity. – If his popularity has not changed, what is the probability that more than half of a sample of 300 voters would vote for him? 23

EXAMPLE (contd. ) Solution • The number of respondents who prefer the representative is binomial with n = 300 and p =. 52. Thus, np = 300(. 52) = 156 and n(1 -p) = 300(1 -. 52) = 144 (both greater than 5) 24