Ch. 7 Stats Vocab

statistic

number that describes some characteristic of a SAMPLE

• x̄ = sample mean

• p̂ = sample proportion

• s_x = sample standard deviation

parameter

the TRUE number that describes some characteristic of the population

• μ = population mean

• p = population proportion

• σ = population standard deviation

• indicate with TRUE or REAL, if you are certain it is the parameter

p̂ or p-hat

number of successes/total sample = x/n

• can sub into the binomal eqns —> μ_x = np & σ = (npq)^1/2

μ_p̂ = mean of sample proportion

p, or the population proportion

σ_p̂ = SD of sample proportion

(pq/n)^1/2

• p = probability success

• q = probability failure

• n = number of trials

• MUST SATISFY THE 10% CONDITION

unlikely or unusual

less than 5%, typically provides convincing evidence against claims

bias in describing samples

where the sample statistics are located

• how accurate

  • high bias = centered around population parameter

  • low bias = centered around different value

variablity in describing samples

how far the sample statistics are located

• how precise

  • high variability = lots of scatter

  • low variability = very little scatter

• reduce by increasing sample size

sampling variability/error

different random samples of same size from same population may produce different values for a statistic

• random chances

sampling distribution

distribution of values taken by the statistics in all possible samples of the same size from the population

• multiple samples combined into one

sampling distribution of the sample proportion

the distribution of values taken by the sample proportion in all possible samples of the same size from the same population

• individual samples’ proportion value combined into a larger data set

shape of a proportion sampling distribution

three possibilities: skewed to left, skewed to right, and symmetric

• n = small, p close to 0 —> skewed to right

• n = small, p close to 1 —> skewed to left

• n = large, p does not matter —> approximately normal

  • can be determined with Large Counts Condition (np ≥ 10, nq ≥ 10)

• n = doesnt matter, p = 0.5 —> normal

center of a proportion sampling distribution

mean of all sample proportions, essentially the population proportion

variability of a proportion sampling distribution

standard deviation of sample proportions, can calculate

• requires the 10% condition

assumptions of sample proportions problems

• random (SRS or stated)

• Independence; 10% Condition (n < 10%N)

• Large Counts (for normal); np ≥ 10 & nq ≥ 10

requirements for describing SD/mean of sampling distributions

• On average

• indicator of type of sampling distribution ex: proportion, mean

• variable differs from the mean by about SD

• sampling size

more extreme with probabilities

more extreme depends on the context of the value

• z-score is on the left (negative side) —> more extreme is from negative infinity to the z-score

• z-score is on the right (positive side) —> more extreme is from z-score to positive infinity

sampling distribution of the sample mean

describes the distribution of values taken by the sample mean in all possible samples of the same size from the same population

μ_x̄ or mean of sampling mean

μ, or the population’s true mean

σ_x̄ = SD of sample mean

σ/(n)^1/2

• must satisfy the 10% condition, n ≤ 10%N

normal distributions of sample means

• if population distribution is normal, then the sampling distribution is too

• if the population distribution is not stated to be normal, use the central limit theorem to create normality

Central Limit Theorem (CLT)

when n is large, the sampling distribution of the sample mean is approximately normal

• n ≤ 30, in most cases

assumptions of sample mean problems

• random (SRS or stated)

• Independence; 10% condition

• approximately normal (stated or n ≤ 30 by CLT)

how to adjust the mean/SD to fit a proportion

1. use inverse norm on the intended proportion

2. plug the z-score back into the equation

3. fill in all other variables, leaving the mean/SD blank (depending on what solving for)

4. complete!