Chapter 7: The Central Limit Theorem

Central limit theorem

If the sample size is large enough then we can assume it has an approximately normal distribution.

Exponential Distribution

a continuous random variable (RV) that appears when we are interested in the intervals of time between some random events

μX

the mean of X

σX

the standard deviation of X

𝑥¯ ~ N(𝜇𝑥, 𝜎𝑋 / 𝑛√)

If you draw random samples of size n, then as n increases, the random variable 𝑥 which consists of sample means, tends to be normally distributed

sampling distribution of the mean

approaches a normal distribution as n, the sample size, increases.

Standard error of the mean

𝜎𝑥 \= 𝜎𝑋 / √𝑛 \=standard deviation of x¯

The central limit theorem for sums

As sample sizes increase, the distribution of means more closely follows the normal distribution.

Central limit theorem formula

∑X ~ N[(n)(μx),(𝑛√n)(σx)]

The normal distribution

has a mean equal to the original mean multiplied by the sample size and a standard deviation equal to the original standard deviation multiplied by the square root of the sample size.

𝑧

𝛴𝑥–(𝑛)(𝜇𝑋) / (√𝑛)(𝜎𝑋)

Law of large numbers

if you take samples of larger and larger size from any population, then the mean x¯ of the sample tends to get closer and closer to μ.

Binomial distribution

there are a certain number n of independent trials. the outcomes of any trial are success or failure. each trial has the same probability of a success p

Σx

is one sum.

(n)(μX)

the mean of ΣX

(√n)(𝜎X)

standard deviation of ΣX

Central limit theorem

If the sample size is large enough then we can assume it has an approximately normal distribution.

Exponential Distribution

a continuous random variable (RV) that appears when we are interested in the intervals of time between some random events

μX

the mean of X

σX

the standard deviation of X

𝑥¯ ~ N(𝜇𝑥, 𝜎𝑋 / 𝑛√)

If you draw random samples of size n, then as n increases, the random variable 𝑥 which consists of sample means, tends to be normally distributed

sampling distribution of the mean

approaches a normal distribution as n, the sample size, increases.

Standard error of the mean

𝜎𝑥 \= 𝜎𝑋 / √𝑛 \=standard deviation of x¯

The central limit theorem for sums

As sample sizes increase, the distribution of means more closely follows the normal distribution.

Central limit theorem formula

∑X ~ N[(n)(μx),(𝑛√n)(σx)]

The normal distribution

has a mean equal to the original mean multiplied by the sample size and a standard deviation equal to the original standard deviation multiplied by the square root of the sample size.

𝑧

𝛴𝑥–(𝑛)(𝜇𝑋) / (√𝑛)(𝜎𝑋)

Law of large numbers

if you take samples of larger and larger size from any population, then the mean x¯ of the sample tends to get closer and closer to μ.

Binomial distribution

there are a certain number n of independent trials. the outcomes of any trial are success or failure. each trial has the same probability of a success p

Σx

is one sum.

(n)(μX)

the mean of ΣX

(√n)(𝜎X)

standard deviation of ΣX