Hp 49g+ User Manual download pdf (Page 592)

100

101

Page 18-25

is the probability of success, then the mean value, or expectation, of X is E[X]

= p, and its variance is Var[X] = p(1-p).

If an experiment involving X is repeated n times, and k successful outcomes

are recorded, then an estimate of p is given by p’= k/n, while the standard

error of p’ is σ

p’

= √(p⋅(1-p)/n) . In practice, the sample estimate for p, i.e., p’

replaces p in the standard error formula.

For a large sample size, n>30, and n⋅p > 5 and n⋅(1-p)>5, the sampling

distribution is very nearly normal. Therefore, the 100(1-α) % central two-sided

confidence interval for the population mean p is (p’+z

⋅σ

p’

, p’+z

⋅σ

p’

For a small sample (n<30), the interval can be estimated as (p’-t

n-1,

⋅σ

p’

,p’+t

⋅σ

p’

Sampling distribution of differences and sums of statistics

Let S

and S

be independent statistics from two populations based on

samples of sizes n

and n

, respectively. Also, let the respective means and

standard errors of the sampling distributions of those statistics be µ

and µ

and σ

, respectively. The differences between the statistics from the

two populations, S

-S

, have a sampling distribution with mean µ

−

= µ

, and standard error σ

−

= (σ

+ σ

)

1/2

. Also, the sum of the statistics

has a mean µ

S1+S2

= µ

+µ

, and standard error σ

S1+S2

= (σ

)

1/2

Estimators for the mean and standard deviation of the difference and sum of

the statistics S

and S

are given by:

2121

SSSS

σσ

σµ +=±=

±±

In these expressions, X

and X

are the values of the statistics S

and S

from

samples taken from the two populations, and σ

and σ

are the variances

of the populations of the statistics S

and S

from which the samples were

taken.

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