CHAPTER 13

• Sampling distributions are introduced

Parameter vs. Statistic

• A population is the entire group of individuals of interest, but it is usually impossible to assess directly
• A parameter is a number that summarizes the population, and parameters are typically unknown
• A sample is a portion of the population that has been examined and provides data
• A statistic is a number that summarizes a sample, used to estimate an unknown population parameter

Sampling Distributions

• Different random samples from the same population yields different statistics, but a predictable pattern emerges over time
• A statistic computed from a random sample is a random variable
• The sampling distribution of a statistic is a probability distribution of the statistic for samples of size n from a given population

Sampling Distribution of the Sample Mean

• The mean of the sampling distribution of x̄ is µ
  • There is no tendency for a sample average to fall systematically above or below µ, even if the population distribution is skewed
• x̄ is an unbiased estimate of the population mean µ
• The standard deviation of the sampling distribution of x̄ is σ/√n
  • The standard deviation measures how much the sample statistic (x̄) varies from sample to sample
• Averages are less variable than individual observations

Normally Distributed Populations

• When a variable in a population is normally distributed, the sampling distribution of the sample mean x̄ is also normally distributed
• For a population, the distribution is N(µ, σ)
• For the sampling distribution, it is N(µ, σ/√n)

Sampling Distribution Example

• Blood cholesterols of 14-year-old boys follow roughly N(µ = 170, σ = 30) mg/dL
• The middle 99.7% of cholesterol levels in boys is 80 to 260 mg/dL.
• In random samples of 25 boys, the sampling distribution of average cholesterol level is roughly N(µ = 170, σ = 30/√25 = 6) mg/dL
• The middle 99.7% of average cholesterol levels (of 25 boys) is 152 to 188 mg/dL

Standardizing Normal Sample Distribution

• If the sampling distribution of the ample mean x̄ is normal, the value of a sample mean x̄ can be standardized into a z-score
• This z-score can be used to find areas under the sampling distribution from Table B
  • x̄ -> z = (x̄ - µ) / (σ /√n) -> z
  • N(µ, σ/ √n) -> N(0,1)
• σ/√n is the standard deviation of its sampling distribution, indicative of spread
• σ is referred to as the standard deviation of the original population

Standardization Example

Hypokalemia is diagnosed when blood potassium levels are low, below 3.5 mEq/dL. Measured potassium levels of a patient vary daily according to N(µ = 3.8, σ = 0.2).

• If one measurement is made, the probability that a patient will be misdiagnosed hypokalemic is around 7%
• Standardize x=3.5 mEq/dL to z=(3.5-3.8)/0.2 = -1.5
• Then P(z < -1.5) = 0.0668 ≈ 7%

Central Limit Theorem

• When randomly sampling from any population with mean m and standard deviation σ, when n is large enough, the sampling distribution of x̄ is approximately normal: N(µ, σ/√n)
• The larger the sample size, the better the approximation of normality
• Many statistical tests assume normality for the sampling distribution, and if the sample size is large enough, we can safely make the assumption even if the raw data appear non-normal

Sample Size

• It depends on the population distribution; if the population distribution is far from normal, more observations are required
• Sample size of 25 or more is enough to obtain a normal sampling distribution from a skewed population, even with mild outliers in the sample
• If the population is extremely skewed and has mild (but not extreme) outliers, a sampe size of 40 or more will suffice

Chapter 12: Proportions

• A population contains a proportion of successes, p
• If the population is much larger than the sample, the count X of successes in a Simple Random Sample (SRS) of size n has approximately the binomial distribution B(n, p) with mean µ and standard deviation σ
  • µ=np
  • σ = √np(1 − p)
• Normal distribution is used to estimate a normal distribution when n is large, and p is not too close to 0 or 1
  • N (μ = np, σ = √np(1 − p)

Sampling Distribution of a Proportion

• When randomly sampling from a population with proportion p of successes, the sampling distribution of the sample proportion p̂ has mean and standard deviation:
  • µp̂ = p
  • σp̂ = √p(1-p)/n
• p̂ is an unbiased estimator of the populatoin proportion p
• Larger samples typically yield close estimates of the population proportion

Normal Approximation

• A sampling distribution for p̂ is never exactly normal, but approximates it as the sample size increases
• The normal approximation is most accurate for any fixed value of n when p is close to 0.5, and least accurate when p is near 0 or 1
• For the normal approximation for p̂
  • N (μ = ρ, σ = √p (1 − p) /n)

Numerical Example

• Color blindness affects 8% of Caucasian American males.
• To calculate the probability of a random sample of 125, where 10% or more are color-blind it is sufficient to use the Normal approximation
  • np = 10 and n(1 – p) = 115
  • N (p = 0.08, √p (1 – p)/n = 0.024)
• z = (p− p) / σ = (0.10 – 0.08) / 0.024 = 0.824
• P(z ≥ 0.82) = 0.2061 (from Table B)

Law of Large Numbers

• The more randomly drawn observations (n) within a sample, the closer the sample mean (x̄) gets to the population mean (m) (quantitative variables)
• The sample proportion (p̂) gets closer to the population proportion (p) (categorical variable)
• The law of large numbers describes what would happen, should samples of increasing size n be taken from a population
• Sampling distribution describes what would happen if all possible samples of a fixed size n were taken
• Both the law of large numbers and sampling distribution are conceptual ideas with mathematical properties; they are not built from data