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 p̂ = Xn 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

P(p̂ < p S.E.(p̂)) = 0.1587 (30)  P(p S.E.(p̂) < p̂ < p + S.E.(p̂)) = 0.683 (31)  P(p̂ > p + S.E.(p̂)) = 0.1587 (32) 

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.
p̂ < p S.E.(p̂)?
2.
p S.E.(p̂) < p̂ < p + S.E.(p̂)?
3.
p̂ > p + S.E.(p̂)?

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?