Statistics 1 - Chapter 06

Critical Value

(Book) In a normal distribution, this is the z score on the borderline separating the z scores likely to occur and those that are unlikely. Typical values are -1.96 & 1.96. (Auto Definition) The value of a statistic required in order to consider the results significant.

Applying Central Limit Theorem to Z Score

(Sample mean minus population mean) divided by (population standard deviation divided by the square root of N) In normal words: Difference between sample mean and population mean divided by the standard error.

z score

A measure of how many standard deviations you are away from the norm (average or mean).

Uniform Distribution

A rare type of population distribution where the population is evenly spread out. Results in a rectangular shape.

Normal Distribution

Bell-shaped curve that results when the values of a trait in a population are plotted against their frequency.

Sampling Distribution of a Statistic

Distribution of all values of the statistic when all possible samples of the same size n are taken from the same population. You plot this by frequency vs sample value.

Sampling Distribution of the Mean

Distribution of sample means, with all samples having the same sample size n taken from the same population. You plot this by frequency vs sample means. Ex: Sampling procedure = roll a die 5 times and find the mean. Population average = 3.5

Sampling Distribution of the Variance

Distribution of sample variances, with all samples having the same sample size n taken from the same population. You plot this by frequency vs sample variances. Ex: Sampling procedure = roll a die 5 times and find the variance, s squared. Population variance = 2.9

Sampling Distribution of the Proportion

Distribution of the sample proportions, with all samples having the same sample size n taken from the same population. You plot this by frequency vs sample proportions. Ex: Sampling procedure = roll a die 5 times and find the proportion of odd numbers. P = 0.5

Normal Quantile Plot

Graph of Points (x,y) where each x value is from the original set of sample data, and each y value is the corresponding z score that is a quantile value expected from the standard normal distribution.

Notation for the Sampling Distribution of x-bar

If all possible random samples of size n are selected from a population with mean μ and standard deviation σ, the mean of the sample means is denoted by μ(sub)x-bar. Also, the standard deviation of the sample means is denoted by σ(sub)x-bar. This is also called the "Standard Error of the Mean".

Rare Event Rule for Inferential Statistics

If the probability of a particular observed event is exceptionally small, such as under 5%, we conclude that the assumption is probably not correct.

Standard Normal Distribution

Normal distribution with a mean of 0 and a standard deviation of 1.

Z Score Notation

P(a < z < b) = denotes the probability that the z score is between a & b. P(z > a) = denotes the probability that the z score is greater than a. P(z < b) = denotes the probability that the z score is less than a.

Distribution of Sample Means

Pop is Normal, Dist is normal, sample mean = pop mean n>30, Dist is normal, sample mean = pop mean n<=30, Dist is not normal, sample mean = pop mean I'm not sure what this is saying.

Area

Region under the curve.

Population Variance Formula

Square root of population standard deviation

Sample Variance Formula

Square root of sample standard deviation

Z score notation

The expression z(sub)α denotes the z score with the area of α to its right.

Central Limit Theorem

The sampling distribution of the mean will approach the normal distribution as N (sample size) increases.

Z score formula

The value minus the mean, divided by the standard deviation. (x - μ) / σ

Percentile

When given a percentile, such as P95, they are asking what value separates the top 5% from the bottom 95%.

Correction for a Finite Population

When sampling w/o replacement and the sample size n is greater than 5% of the finite population size N, adjust the standard deviation of sample means σ(sub)x-bar by multiplying it by the finite population correction factor.

Continuity Correction

When we use the normal distribution as an approximation to the binomial distribution, a continuity correction is made to a discrete whole number x in the binomial distribution by representing the discrete whole number x by the interval from x - 0.5 to x + 0.5.

Purpose behind "Continuity Correction"

When you use a normal distribution with discrete data and need to find a success % for a specific value.. If your value is 2000 out of 2500 people for whatever measured trait, it's n=2500 & x=2000. p=0.8 & q=0.2. Just figure out the z score of 2499.5 & 2500.5. Find the success between those two values, and ta-da!

Sample Standard Deviation Formula

sum of [ (sample value minus sample average) squared] divided by numbers of samples less one.

Population Standard Deviation Formula

sum of [ (value minus population average) squared] divided by population number