Applied Statistics - Chapter 5: Probability & the Standard Normal Distribution + Sampling Distribution of Means

Vocabulary practice flashcards covering probability concepts, terminology, formulas, standard normal distributions, Z-score transformations, and percentile ranks based on the lecture notes.

Inferential Statistics

A branch of statistics that uses sample data to answer questions about an entire population, defining relationships between samples and populations in terms of probability.

Probability

Reflects the likelihood of an event occurring,

Probability (con’t)

It’s calculated in its simplest form as:

• The number of outcomes of interest divided by The total number of possible outcomes.

Probability’s purpose

To estimate attributes of the population given information that you know about the sample

Sample Space

A set that contains all possible outcomes of a given experiment or random process.

e.g., A die has six possible outcomes; therefore, the sample space contains those six outcomes.

Event

A particular outcome or set of outcomes in the sample space, forming a subset of it

e.g. Flipping a coin once – an event might be “heads”

Probability Notation $p(A)$

The notation for the probability of event $A$ occurring, with values ranging between $0$ and $1$ ($0 < p(A) < 1$)

e.g. If you flip a fair coin, what is the probability of getting a head? p(head) = 1/2 = .50

What can probability Notation p(A) be expressed as?

It can be expressed as a proportion, fraction, or ratio.

Probability for Continuous Variables

The probability calculated for continuous variables, defined as $p = \text{area under curve for a range of scores}$.

Standard normal distribution

When the distribution of scores (X) is normal

Standard normal distribution (con’t)

We can convert X scores to Z scores and use the standard normal distribution

What does the standard normal distribution allow us to do

It allows us to determine the proportion of area within specific areas of the normal distribution

How many standard deviations are -1/+1 away from the mean

68.26%

How many standard deviations are -2/+2 away from the mean

95.44%

How many standard deviations are -3/+3 away from the mean

99.74%

Column B (Standard Normal Table)

The column in Appendix A that represents the proportion of area in the body (the total area before $Z$).

Column C (Standard Normal Table)

The column in Appendix A that represents the proportion of area in the tail (the area beyond $Z$).

Column D (Standard Normal Table)

The column in Appendix A that represents the proportion of the area located between the mean ($Z = 0$) and a specific Z-score.

Percentile Rank

An indicator that shows a specific score is greater than or equal to a given percentage of scores within a distribution.

Extreme $5\text{\%}$ Cutoff Z-Scores

The Z-scores ($Z = +1.96$ and $Z = -1.96$) associated with the most extreme $5\text{\%}$ of scores in a normal distribution, dividing the extreme region into $.0250$ in each tail.

Negative Z scores

The percentile rank of a score below the mean

How can negative Z scores be found?

• Column C (area beyond Z score) x 100

• [.50 – Column D (area between mean and Z score)] x 100

Positive Z scores

The percentile rank of a score above the mean

How can positive Z scores be found?

• [1 – Column C (area beyond Z score)] x 100

• [.50 + Column D (area between mean and Z score)] x 100

Sampling distribution

A theoretical probability distribution of the possible values of some sample statistic that occurs if we’re to collect an infinite number of same-sized samples from a population

i.e. A frequency distribution, except that it is for a sample statistic such as M or rxy instead of X

Sampling distribution (con’t)

There’s a sampling distribution for every different sample statistic

e.g. Mean (M), variance, standard deviation, linear correlation, etc

Sampling distribution (III)

The sample means pile up around the population mean, and the distribution of sample means is approximately normal in shape.

The purpose of sampling distribution

To help us interpret the value of the sample statistic.

Sampling error

The random difference (amount of error) between a sample statistic and the population parameter

Sampling error (con’t)

It's the natural discrepancy, or error, between a sample statistic and the population parameter it estimates, with varying samples

i.e. Doesn’t indicate a mistake was made.

When is the distribution of sample means almost perfectly normal?

When the population from which the samples are selected is a normal distribution

When is the distribution of sample means almost perfectly normal? (con’t)

When the number of scores (n) in each sample is relatively large, around 30 or more

Central limit theorem

When the distribution of sample means approaches normality as the sample size (N) increases, regardless of the shape of the population distribution of X

Central limit theorem (con’t)

As N becomes larger (approaching infinity), the sampling distribution of the mean (M) becomes more normally distributed

The assumptions of central limit theorem

We assume that the population has a mean and a standard deviation that are fixed

The assumptions of central limit theorem (con’t)

We assume that our sampling is random (each unit in the population has an equal probability of being selected)

The assumptions of central limit theorem (III)

We assume that each sample will have the same number of observations

What factors does standard error depend on?

1. Standard deviation in the population

2. Sample size

Law of large numbers

The larger the sample size, the more probable it is that the sample mean will be close to the population mean.

The thing about population variance

The greater the variance in the population, the less probable it is that the sample mean will be close to the population mean.

The main usage of the distribution of sample means

It’s mainly used to find the probability associated with any particular sample

The thing about distribution of sample means

Proportions of the normal curve are used to represent probabilities.

Distribution of sample means and Z-scores

A z-score for sample means is computed.

Statistical inference

A series of procedures in which data obtained from samples are used to make statements about a broader set of circumstances or populations

Statistical inference (con’t)

Uses sample data to draw general conclusions about populations, but the sample isn’t perfectly accurate reflection of its population

Statistical inference (III)

Differences between sample and population introduce uncertainty into inferential processes

Statistical inference (IV)

Its techniques use probabilities to draw inferences from sample data

Point estimation

The use of a sample statistic to estimate the value of a population parameter

Interval estimation

The designation of a range of values that has a specific probability of encompassing the value of the population parameter.

Interval estimation (con’t)

Is also the estimation of the amount of variability in a sample statistic when many samples are taken repeatedly from the same population

Hypothesis Testing

A comparison of the value of a sample statistic with the hypothesized value of a population parameter.