Parameters, Statistics and Sampling Distributions

Parameter versus Statistic

• Population refers to the entire group of individuals of interest, but it is usually impossible to assess directly.
• A parameter is a number summarizing the population and is usually unknown.
• A sample is a portion of the population examined, providing data for analysis.
• A statistic is a number summarizing a sample and are used to estimate population parameters.

Sampling Distributions

• Different random samples yield different statistics, but there is a predictable pattern in the long run.
• A statistic computed from a random sample is a random variable.
• The sampling distribution of a statistic is its probability distribution for samples of a given size n taken from a given population.

Sampling Distribution of the Sample Mean

• The mean of the sampling distribution is μ.
• There's no systematic tendency for a sample average to fall above or below μ, even with a skewed population distribution ensuring that this is an unbiased estimate of the population mean μ.
• The standard deviation of the sampling distribution is σ/√n.
• The standard deviation of the sampling distribution measures the variation of the sample statistic from sample to sample.
• Averages variable less than individual observations.

Normally Distributed Populations

• For a variable in a population that is normally distributed, the sampling distribution of the sample mean x bar is also normally distributed.

Sampling Distribution Example: Cholesterol Levels

• Blood cholesterol levels of 14-year-old boys: ~ N(μ = 170, σ = 30) mg/dL.
• The middle 99.7% of cholesterol levels in boys is 80 to 260 mg/dL.
• Now consider random samples of 25 boys, the sampling distribution of average cholesterol levels is ~ 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.

Another Sampling Distribution Example : Deer mice

• Deer mice (Peromyscus maniculatus) vary normally in body length with a mean body length μ = 86 mm, and standard deviation σ = 8 mm.
• For random samples of 20 deer mice, the best answer is: Normal, mean 86, standard daeviation 1.789 mm.

Standardizing a Normal Sample Distribution

• For a normal sampling distribution, standardize the value of a sample mean x bar to get a z-score, used to find areas under the sampling distribution from Table B.
• When working with the sampling distribution, σ/√n is its standard deviation, indicating spread.
• σ is the standard deviation of the original population.

Standardization Example: Hypokalemia Diagnosis

• Hypokalemia diagnosis: blood potassium levels below 3.5 mEq/dL. Assume measured potassium levels vary daily according to N(μ = 3.8, σ = 0.2).
• If only one measurement is made, the probability that the patient will be misdiagnosed hypokalemic is P(z < 1.5) = 0.0668 ≈ 7%.
• Taking measurements over four separate days affects the probability of misdiagnosis.
• Averaging 4 measurements is more likely to be closer to the true average than individual measurements.

The 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 x bar is approximately Normal: N(μ, σ/√n).
• The larger the sample size n, the better the approximation of Normality.
• The CLT is useful in inference: Many statistical tests assume Normality for the sampling distribution; if the sample size is large enough, this assumption is safe even if raw data appears non-Normal.

How Large a Sample Size?

• It depends on the population distribution; more observations are required when the population distribution deviates from normality.
• A sample size ≥ 25 is enough to obtain a normal sampling distribution from a skewed population, even one containing mild outliers.
• A sample size of 40 or more should overcome an extremely skewed population and mild, but not extreme, outliers.
• In many cases, n =25 is not a large sample

When the Population Is Skewed

• The sampling distribution is approximately normal, assuming the central limit theorem, even though the population is strongly skewed.

Is the Population Normal?

• Variables sometimes have an approximately normal distribution, however most of the time we only have sample data.
• Summarize data with a histogram and describe its shape.
• If the sample is random, the histogram's shape should be similar to the population distribution.
• The central limit theorem can help guess whether the sampling distribution should look roughly normal.

Central Limit Theorem Examples

• Angle of big toe deformations in 38 patients is symmetrical and has one small outlier: the population is likely close to normal, thus the sampling distribution is ~ normal.
• Histogram of servings of fruit per day for 74 adolescent girls: Skewed, no outlier. The population is likely skewed, but the sampling distribution ~ Normal given large sample size.
• Atlantic acorn sizes (in cm³) with a sample of 28 acorns.
• Describe the distribution of the sample; what can one assume about the population distribution?
• What would the shape of the sampling distribution be for samples of sizes 5, 15 and 50?

Chapter 12: Proportions

• A population contains a proportion p of successes.
• 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 u and standard deviation σ.
• If n is large, and p is not too close to 0 or 1, this binomial distribution can be approximated by the Normal distribution.

Sampling Distribution of a Proportion

• With random sampling, the proportion p of successes, the sampling distribution of the sample proportion ["p hat"] has mean and standard deviation.
• The sample proportion is an unbiased estimator of the population proportion p.
• Larger samples usually give closer estimates of the population proportion p.

Normal Approximation

• The sampling distribution of p is never exactly Normal.
• As the sample size increases, the sampling distribution of p^ becomes approximately normal.
• Normal approximation is most accurate for any fixed n when p is close to 0.5, and least accurate when p is near 0 or 1.
• When n is large, and p is not too close to 0 or 1, the sampling distribution of p^ is approximately:

Numerical Example

• The frequency of color blindness (dyschromatopsia) in the Caucasian American male population is about 8%.
• Suppose a random sample of size 125 from this population, what is the probability that ≥10% of the sample are color-blind?
• A sample size of 125 is large enough to use of the Normal approximation (np = 10 and n(1 – p) = 115).

Normal approximation for p^ sampling distribution:

• z = ( − p) / σ = (0.10 – 0.08) / 0.024 = 0.824
• P(z ≥ 0.82) = 0.2061 from Table B
• Or P(p ≥ 0.10) = 1 – NORM.DIST(0.10, 0.08, 0.024, 1) = 0.2023 (Excel) = normalcdf (0.10, 1E99, 0.08, 0.024) = 0.2023 (TI-83)

The Law of Large Numbers

• As the number of randomly drawn observations (n) in a sample increases, the mean of the sample () gets closer and closer to the population mean m (quantitative variable).
• As the number of randomly drawn observations (n) in a sample increases, the sample proportion () gets closer and closer to the population proportion p (categorical variable).
• Note: When sampling randomly from a given population,
  • The law of large numbers describes what would happen if we took samples of increasing size n.
  • A sampling distribution describes what would happen if we took all possible random samples of a fixed size n. Both are conceptual ideas with many important practical applications.
  • Rely on their known mathematical properties, but we don't actually build them from data.

Understand the difference between population parameters and sample statistics. Learn about sampling distributions and how sample means estimate population means. Explore the properties of sampling distributions of the mean.