Statistikk og Simulering

Økt 13. Estimation of the Mean

The theoretical distribution

14.1. The theoretical distribution

For large $n$ the binomial distribution is approximately equal to the normal distribution, and hence $\stackrel{̂}{p}=X∕n$ has normal distribution. This fact follows from the Central Limit Theorem. Unless $p$ is very close to 0 or 1, $n>25$ qualifies as large. For a normal distribution, we have

You can find these probabilities in a table for the standard normal distribution (or using cdf in Matlab).

Oppgåve 14.1 Review Exercise 12.4 where you counted three events:

1.
$\stackrel{̂}{p}?
2.
$p-S.E.\left(\stackrel{̂}{p}\right)<\stackrel{̂}{p}?
3.
$\stackrel{̂}{p}>p+S.E.\left(\stackrel{̂}{p}\right)$?

Use Equations (30)–(32) and find the expected number of occurrences of each of these events.

Øving 14.1 Discuss:

What is the relationship between the sample mean, the population mean, and the expected value?