I have a test that is going to be on Confidence Interval Lar

I have a test that is going to be on Confidence Interval, Large Samples, Small Samples, Paramater, and Proportion Percentage.

I am having trouble writing a formula sheet for the test. Can someone help me?

Test that is going to be on confidence interval , Large samples.parameter and proportion percentage.

Parameter :

For large samples and condition for this is np > 5 and nq > 5.

n = Sample size. (n > = 30)

p ^ = sample statistic = r / n

Where , r = number of successes.

p₀ = Proportion specified in null hypothesis.

q₀ = 1 - p

The distribution of p^ is approximately Normal with mean = µ =p

and standard deviation = = sqrt ( p₀ *q₀ /n )

We use the test statistic,

Z = (p^ - p₀ ) / sqrt( (p₀ *q₀ )/n ).

A confidence interval for p,

p^ - E < p < p^ + E

E = Zc * sqrt ((p₀ * q₀ )/n)
Zc is critical value for confidence interval c.

b) Test for differences of proportions:

i) We will draw independent random samples of size n1 and n2 from two binomial distributions.

ii) For large values of n1 and n2 the distribution of sample differences in proportion of successes will be approximately Normally distributed.

iii) r1 and r2 are number of successes from the first and second sample.

iv) p1 and p2 are probability of successes on each trial from the first and second binomial distributions.

For large values of n1 and n2 the distribution of sample differences,

p1^ -p2^ = (r1 / n1) - (r2 / n2)is approximately normal with mean = p1 - p2

and standard deviation = sqrt ( (p1*q1)/n1 + (p2*q2)/n2)

Where, q1 = 1 - p1 and q2 = 1 -p2

If the proportion p1 and p2 are unknown and must be estimated.

Best pooled estimate = p^ = ( r1 + r2 ) / (n1 + n2) and q^ = 1 - p^

Large sample condition :

n1*p^ , n2*p^ , n1*q^ and n2*q^ all are >5.

The test statistic for this is ,

Z = (p1^ - p2^) / sqrt (( p^*q^)/n1 + (p^*q^)/n2)

The confidence interval for difference of proportion is ,

(p1^ - p2^) - E < p1 - p2 < (p1^ - p2^) + E

Where E = Zc * sqrt (( p^*q^)/n1 + (p^*q^)/n2)

These are the formula sheet for the test.