Module 5 Notes

Página 1

The normal approximation to the binomial can be very accurate if n is relatively

large and p is close to 0.5. The condition with respect to p is reasonable

because the binomial distribution is symmetric for p = 0.5, so the shape of the

binomial distribution is closer to the normal symmetric shape as p approaches

0.5. Recall that for a binomial distribution with parameters n and p, the mean =

np and the variance = np(1 – p).

When using the normal distribution to approximate the

binomial, the mean and the variance are calculated as:

µ = np

σ2 = np (1 – p)

The general rule is that the approximation of the binomial by the normal

distribution is good if both np and n(1 – p) are equal to or greater than five.

confidence levels

To determine the z-value associated with a 95% confidence interval in the normal curve table, you first divide the error in half and write it as /2. For = 0.05, /2 = 0.025. You then look up the z-value in the table associated with an area under the curve of 0.975 (1.000 – 0.025), which is 1.96. Alternatively, you can look up the z-value associated with 0.025, which is –1.96.sampling distribution of the proportion:

You multiply the standard error by the finite population correction factor

(Topic 4.4) when all of the following are true:

The size of the population is known.

The sample size is greater than 5% of the population size.

The sampling occurs without replacement.

Confidence interval for the population mean (σ unknown)

In most cases, you will not know the population standard deviation. If the

value of σ is unknown, you substitute the value for the sample standard

deviation in the formula for the standard error. This substitution can be safely

done when the sample size exceeds 30.

You also use the t distribution to construct the confidence interval when the

value of σ is unknown and the sample size is less than or equal to 30.

Confidence interval for the population proportion Recall from Topics 3.4 and 5.1 that the population proportion, p, is a binomial

parameter. Yet the confidence interval formula for the population proportion

uses the normal probability distribution to approximate the binomial

distribution. This use of the normal curve approximation is valid and fairly

accurate as long as the sample size is sufficiently large; the rule is that both

np and n(1 – p) must be equal to or greater than five.