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Statistics

Notes 2. Using Central Limit Theorem, the above result will also hold for a non-normal population
when both n and n > 30 and fpc is approximately equal to unity, i.e., n < 0.05 N
1 2 i i
(for i = 1, 2).
24.3 Sampling Distribution of the Number of Successes

Let p denote the proportion of successes in population, i.e.,

Number of successes in population

p =

Total number of units in population

Let us take a random sample of n units from this population and let X denote the number of
successes in the sample. Thus, X is a random variable with mean np and standard error
æ N n ö
-
np (1 p- ) or × np (1 p- ) ÷
ç
-
è N 1 ø
If sampling is done with replacement, then X is a binomial variate with mean np and standard
error np (1 p- ) . Using central limit theorem, we can say that the distribution of the number
of successes will approach a normal variate with mean np and standard error np (1 p- ) or
N n
-
× np (1 p- ) for sufficiently large sample. The sample size is said to be sufficiently large
N 1
-
if both n p and n(1 - p) are greater than 5.
24.3.1 Sampling Distribution of Proportion of Successes

X
Let p = be the proportion of successes in sample. Since X is a random variable, therefore, p
n
is also a random variable with mean

( )
E X np
E ( ) p = = = p and standard error
n n

1 np (1 p- ) p (1 p- )
( ) =
= 2 Var X 2 = (when srswr)
n n n
N n p (1 p- )
-
or = × (when srswor)
N 1 n
-
As in the previous section, the sampling distribution of p will also be normal if both n p and
n(1 - p) are greater than 5.

Example 2: There are 500 mangoes in a basket out of which 80 are defective. If obtaining
a defective mango is termed as a success, determine the mean and standard error of the proportion
of successes in a random sample of 10 mangoes, drawn (a) with replacement and (b) without
replacement.

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