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Example 2.2 A survey of 313 children, ages 14 to 22, selected from the nation’s top corporate executives; when asked to identify the best aspect of being privileged in this group, 55% mentioned material and financial gains. a) describe the sampling distribution of the sample proportion b) assume that the population proportion is 0.5; what is the probability of ˆ? observing a sample proportion as large or larger than p

Solution

ˆ is normally distributed a) since the sample size is large, then the distribution of p ˆ = 0.55 and σ p with mean p ˆ =
pq ≈ n ˆˆ pq 0.55 × 0.45 = = 0.028 313 n

ˆ will fall within therefore, we know that approximately 95% of the time p 2σ p ˆ ≈ 0.056 of the unknown value of p . ˆ ; ie. One could check the condition that allows for normal approximation to the distribution of p ˆ ± 2σ p p ˆ = 0.55 ± 0.056 or 0.494 to 0.606, which falls in the interval 0 to 1
pq 0.5 × 0.5 = = 0.0283 n 313

b) we are given that µ p ˆ = p = 0.5 and σ p ˆ =

The sampling distribution of based on a sample of children

0.55 − 0.5   ˆ ≥ 0.55 ) = P  Z ≥ P( p  0.0283  

⇒ P ( Z ≥ 1.77 ) = 0.0384

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This tells us that if we were to select a random sample of n = 313 observations from a ˆ would be as population with proportion p = 0.5 , the probability that the sample proportion p large or larger than 0.55 is only 4%. Alternatively: using the correction of continuity, the equivalent to ±0.5 would be ± 21n , So;

 ( 0.55 − 0.0016 ) − 0.5  = P Z ≥ 1.71 = 0.0436 P  Z1 ≥ ( )  0.0283   When n is large, the effect of using the correction is generally negligible.

2.6

Sampling distribution: Sum or Difference between two sample mean When independent random samples of size n1 and n2 observations have been
2 selected from population with means µ1 and µ2 , and variances σ 12 and σ 2

respectively; the sampling distribution of the sum or differences will have the following properties: (a) The mean and standard deviation of ( x1 ± x2 ) :

µ( x1 ± x2 ) = µ1 ± µ 2

and

σ ( x1 ± x2 ) =

σ 12
n1

+

2 σ2

n2

(b) If the sampled populations are normally distributed, then the sampling distribution is exactly normally distributed regardless of the sample size (c) If the sampled populations are not normally distributed, then the sampling distribution is approximately normally distributed when the sample size are large due to the CLT

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