PP

Chapter 7: The Central Limit Theorem

Introductory

  • 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

7.1 The Central Limit Theorem for Sample Means (Averages)

  • 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.

  • 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.

    Central Limit Theorem for Sample Means z-score and standard error of the mean

  • 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 distribution
    • sample 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 kth percentile
    • mean is the mean of the original distribution
    • standard deviation is the standard deviation of the original distribution
    • sample size = n

7.2 The Central Limit Theorem for Sums

  • 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 ΣX has the following z-score associated with it:

    • Σx is one sum.
    • 𝑧 = 𝛴𝑥–(𝑛)(𝜇𝑋) / (√𝑛)(𝜎𝑋)
    1. (n)(μX) = the mean of ΣX
    2. (√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 kth percentile
    • mean is the mean of the original distribution
    • standard deviation is the standard deviation of the original distribution
    • sample size = n

7.3 Using the Central Limit Theorem

  • 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 successful p

Examples