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.The sample size has to be greater than 30 to assume 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¯ ~****(𝜇𝑥, 𝜎𝑋 / 𝑛√)** = If you draw random samples of size*N**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.The random variable 𝑥¯ has a different

*z*-score associated with it from that of the random variable*X*. The mean 𝑥¯ is the value of 𝑥¯ in one sample.

*μX*= the average of both*X*and x¯**Standard error of the mean**: 𝜎𝑥 = 𝜎𝑋 / √𝑛 =standard deviation of x¯Probabilities for means on the calculator

2nd DISTR

2:normalcdf

normalcdf (lower value of the area, upper value of the area, mean, standard deviation / √sample size)

where

*mean*is the mean of the original distribution*standard deviation*is the standard deviation of the original distributionsample size = n

Percentiles for means on the calculator

2nd DISTR

3:InvNorm

*k*= invNorm (area to the left of 𝑘, mean, standard deviation / √sample size)Where→

*k*= the*k*th percentile*mean*is the mean of the original distribution*standard deviation*is the standard deviation of the original distribution*sample size*= n

**The****central limit theorem for sums:**As sample sizes increase, the distribution of means more closely follows the normal distribution.*∑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**.**The random variable Σ

*X*has the following*z*-score associated with it:Σ

*x*is one sum.𝑧 = 𝛴𝑥–(𝑛)(𝜇𝑋) / (√𝑛)(𝜎𝑋)

(

*n*)(*μX*) = the mean of Σ*X*(√n)(𝜎X) = standard deviation of ΣX

Probabilities for sums on the calculator

2nd DISTR

2: normalcdf (lower value of the area, upper value of the area, (

*n*)(mean), (√n)(standard deviation))where:

*mean*is the mean of the original distribution*standard deviation*is the standard deviation of the original distribution*sample size*= n

Percentiles for sums on the calculator

2nd DIStR

3:invNorm

*k*= invNorm (area to the left of*k*, (*n*)(mean), (√n)(standard deviation)where:

*k*is the*k*th percentile*mean*is the mean of the original distribution*standard deviation*is the standard deviation of the original distribution*sample size*= n

**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 trialsthe outcomes of any trial are success or failure

each trial has the same probability of a successful

*p*

**Central limit theorem**: If the sample size is large enough then we can assume it has an approximately normal distribution.The sample size has to be greater than 30 to assume 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¯ ~****(𝜇𝑥, 𝜎𝑋 / 𝑛√)** = If you draw random samples of size*N**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.The random variable 𝑥¯ has a different

*z*-score associated with it from that of the random variable*X*. The mean 𝑥¯ is the value of 𝑥¯ in one sample.

*μX*= the average of both*X*and x¯**Standard error of the mean**: 𝜎𝑥 = 𝜎𝑋 / √𝑛 =standard deviation of x¯Probabilities for means on the calculator

2nd DISTR

2:normalcdf

normalcdf (lower value of the area, upper value of the area, mean, standard deviation / √sample size)

where

*mean*is the mean of the original distribution*standard deviation*is the standard deviation of the original distributionsample size = n

Percentiles for means on the calculator

2nd DISTR

3:InvNorm

*k*= invNorm (area to the left of 𝑘, mean, standard deviation / √sample size)Where→

*k*= the*k*th percentile*mean*is the mean of the original distribution*standard deviation*is the standard deviation of the original distribution*sample size*= n

**The****central limit theorem for sums:**As sample sizes increase, the distribution of means more closely follows the normal distribution.*∑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**.**The random variable Σ

*X*has the following*z*-score associated with it:Σ

*x*is one sum.𝑧 = 𝛴𝑥–(𝑛)(𝜇𝑋) / (√𝑛)(𝜎𝑋)

(

*n*)(*μX*) = the mean of Σ*X*(√n)(𝜎X) = standard deviation of ΣX

Probabilities for sums on the calculator

2nd DISTR

2: normalcdf (lower value of the area, upper value of the area, (

*n*)(mean), (√n)(standard deviation))where:

*mean*is the mean of the original distribution*standard deviation*is the standard deviation of the original distribution*sample size*= n

Percentiles for sums on the calculator

2nd DIStR

3:invNorm

*k*= invNorm (area to the left of*k*, (*n*)(mean), (√n)(standard deviation)where:

*k*is the*k*th percentile*mean*is the mean of the original distribution*standard deviation*is the standard deviation of the original distribution*sample size*= n

**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 trialsthe outcomes of any trial are success or failure

each trial has the same probability of a successful

*p*