In the Name of Allah, the most Gracious,the most Merciful

M.SHAHAB YASEEN

BSSS-13-15

Presentation Topic

Estimation by Confidence Interval(Confidence interval estimation of a population Mean)

Contents Estimationi. Point Estimationii. Interval Estimation Confidence Interval estimate of a population Meani. Normal population with σ knownii. Normal population with σ unknowniii. Non-normal population with known or unknown σ Confidence Interval for Difference of Meansi. Normal population with known Standard Deviationii. Normal population with unknown Standard Deviationiii. Non-normal populations

EstimationIt is a procedure of making judgment about the true but unknown values of the population parameters from the sample observationsIt is further divided into two parts

Point Estimation

Interval Estimation

Confidence IntervalThe concept of confidence interval was introduced in 1937 by the polish-English-American statistician Jerzy NeymanA confidence interval is an interval constructed from the sample values in such a way that it has a known probability such as 95% or 99% etc. of containing some parameter θ intended to be estimate.Suppose θ is a parameter and L,U are two quantities derived from the sample P(L< θ<U)= 1 0< <1 Where is the level of significance The probability 1- , associated with an interval estimate is called

confidence co-efficient or confidence level.

The bounds L,U are called lower and upper confidence limits and the interval L< θ<U is called 100(1- )% confidence interval of θ

For example if =0.05 then the confidence interval is 95% confidence interval

The difference between U-L is called the precision of the confidence interval.it can be increased by increasing the sample size.

Confidence Interval estimate of a population Mean

To compute a confidence interval for the population mean µ,we have to see:i. Weather or not the population is normalii. Weather or not the population standard division is knowniii. Weather the sample size is small or largeWe discuss these different cases below.

1. Normal population with σ known

Suppose that a random sample , …., of size n is drawn from a normal population unknown mean µ and known standard deviation σ and we drive to construct confidence interval for the population mean µ as estimate of which is provided by the sample mean X bar. From the sampling distribution of the mean we know that variable X bar has mean µ and S.D = so that statistic

Z= The normal dis. Tells us that the prob. That a value of Z will fall in

the interval from to is equal to 1

That is we can make the following prob. statementP[ < Z< ] = 1 P[ < < ] = 1 [Z=]Multiply all the terms inside the bracket by and getP[ << ] = 1 By subtracting X bar from each term we haveP[ << + ] = 1

We multiply all terms by -1P[+ > ] = 1 3< 2 if multiply 1 then 3>2]By rearranging we get P[< + ] = 1 Hence a100(1 )% confidence interval for µ is [, + ]Which may be expressed more efficiently as

Suppose we desire a 95% confidence interval,i.e.0.95 100 % C.I(1 0.05 )100 % C.I=0.05 and =0.025From Area table we seeZ=1.96So 95% C.I for Is 1.96< < +1.96 Similarly for 99% we have Z0,005=2.58 2.58<< + 2.58

Example 15.18 Solution:- =1.8 ml , n=8, find 90% C.I X=481,479,482,480,477,478,481,482==1.645=The 90% C.I for is 480 1.645 ()480 1.645 (0.636)480 1.05478.95< <481.05 90% C.I for is (478.95,481.05)

Normal population with σ unknown

If n is sufficiently large (i.e. n) and σ is unknown, the sampling distribution of will be approximately normal with mean µ and S.D= (where S is the sample S.D) hence (1 )100 % C.I for µ is Note: If σ is unknown and the sample size n<30 then the sampling distribution of will not be normal but it has a students T-distribution where s=

Example 15.20Solution: here =5410 ,S=680 ,n=64Because of 90 % confidence interval we have=1.645hence (1 )100 % C.I for µ is 5410 1.645 5410 1.645 (85)5410 139.85270.2 < < 5549.8Hence 90% C.I for is (5270,5550)

Non-normal population with known and unknown σ (large sample)

(1 )100 % C.I for µ,mean of a non-normal population with known is given by When is unknown When sample size n is greater than 5% of population size N then the C.I estimate for µ is

Confidence Interval for Difference of Means

To construct the C.I for the diff. between two means ,the following three cases are to be considered Both the populations are normal with known S.Ds Both the populations are normal with unknown S.Ds Both the populations are non-normal, in which case, both

sample sizes are necessarily large

Normal population with known S.Ds

Suppose we have two normal population having unknown means and and known S.D and .suppose independent Random of size and are drawn from the population respectively and let , represent the sample means then the sampling distribution of d= (i.e. the diff. between means) will be normal with mean and S.D=

Z= is exactly normal, no matter how the sample sizes are.We can therefore make the following prob. Distribution

P[<<]= 1 Multiply each term in the bracket by P[<]= 1 Subtracting to each term inside the bracket, we getP[< ]= 1 Now multiplying each term by 1

P[ ]= 1 OrP[< ]= 1

Hence 100(1 )% C.I for ) is

Normal population with unknown S.Ds

If prob. S.D is unknown, but sample size n and is replaced with and

Non-normal population with known and unknown S.Ds

An Approximate 100(1 )% C.I for ),When the population S.Ds are known then An Approximate 100(1 )% C.I for ),When the population S.Ds are unknown then

Thank YouAny Question?