Lecture Exam #2 Study - CJ Stats.

Know the normal distribution including the relationship between standard deviations

(-/+3, -/+2, -/+1) and percentages (68%, 96%, 99%), and z-score (e.g., -3, -2, -1 0, 1, 2, 3)

Probability of Failure

(Opposite of success is failure) When dealing with event of two potential outcome, we need to know the probability of failure in addition to that of success; Represented by q

Probability of Success

(symbolized p) This is obtained on the basis of prior knowledge or theory. e.g; We know that 62% of felony defendants obtain pretrial release - this means that each defendant's probability of release is p=.62

99% confident level - to gain precision we will..

→ Counter the lack of precision and wide confidence interval - make the sample size larger

Addition Rules

* Can be extended to cover more than one event * If events A and B are mutually exclusive, the probability of getting either event A or B is equal to the probability of event A plus the probability of event B P(a or b)=(a)to(b)

What is a failure vs. a success?

-. Success is the outcome os interest in a trial - Failure is any outcome other than success or the event of interest

What is the difference between independence and mutually exclusive?

-> A mutually exclusive event can simply be defined as a situation when two events cannot occur at same time -> Independent event occurs when one event remains unaffected by the occurrence of the other event.

"with replacement" vs "without replacement"

-with replacement- the item chosen is returned to the original group, denominator stays the same -without replacement-the item will not be returned, number of possible outcomes will change to reflect that

Probability of an event must be between

0 and 1 ; CANNOT BE zero with rounding, must be closer to zero

Example to understand the formula

1. Coin Flips: → any two sided coin will land on heads or tails: a. Denominator in the probability formula is 2 b. Numerator is 1 because there is one side of tail on the coin = p(A) = 1/2 ⇒ .50 (anytime you flip a coin, there is a .50 probability that the coin will land on tails, same goes for heads) * Probabilities together are = p(tails) + p(heads) = .50 + .50 ⇒ 1.00 (probability sum is 1.00)

The Z Distribution offers three pieces of information

1. How close a score is to the distribution mean - Calculate Z-scores of 0: it is greater to the mean of the distribution 2. Whether a score is greater than or less than the mean - Positive number is greater than the mean, negative number is less than the mean 3. A score relative distribution across the population

z score conveys two pieces of information about the raw score on which the z score is based

1. The absolute value of the z score reveals the location of the raw score in relation to the distribution mean. * Z scores are expressed in standard deviation units. A z score of .25, for example, tells you that the underlying raw score is one-fourth of one standard deviation away from the mean. A z score of -1.50, likewise, signifies a raw score that is one-and-one-half standard deviations away from the mean. 2. The second piece of information is given by the sign of the z score. Although standard deviations are always positive, z scores CAN BE NEGATIVE. A z score's sign indicates whether the raw score that the z score represents is greater than the mean (producing a positive z score) or is less than the mean(producing a negative z score) ⭕️ A z score of .25 is above the mean, while a score of -1.50 is BELOW the mean

Two types of probability distributions

1. binomial 2. continuous

What is the Central Limit Theorem?

1. the random sampling distribution of means will always tend to be normal, irrespective of the shape of the population distribution from which the samples were drawn. 2. The random sampling distribution of means will become closer to normal as the size of the sample increases 3. the mean of the random sampling distribution of means is equal to the mean of the original population -> states that if you have a population with mean μ and standard deviation σ and take sufficiently large random samples from the population with replacement , then the distribution of the sample means will be approximately normally distributed.

Normal Curve

A distribution of raw scores from a sample or population that is symmetric and unimodal and has an area of 1.00. Normal curves are expressed in raw units and differ from one another in metrics, means, and standard deviations.

What is the "margin of error" and how does it relate to confidence intervals?

A margin of error tells you how many percentage points your results will differ from the real population value. For example, a 95% confidence interval with a 4 percent margin of error means that your statistic will be within 4 percentage points of the real population value 95% of the time.

Binomial probability distribution

A numerical or graphical display showing the probability associated with each possible outcome of a trial.

Binomial

A trial with two possible outcomes. Also called a dichotomous or binary variable empirical outcome.

For example, if you select a 60% confidence level, how much error are you accepting?

Accepting error rate of 40%

Trial

An act that has several different possible outcomes

Population distribution

An empirical distribution made of raw scores from a population (contains all values in the entire population)

Importance of Probability

Because data used in statistical analyses often involves some amount of "chance" or random variation, understanding probability helps us to understand statistics and how to apply it.

How to use percentile rank to find a z-score?

Find the z-score number on the sides

How to use the z-table to find a percentile rank?

Go to the z-score table to find the closest rank

Representativeness:

How closely the characteristics of a sample match those of the population from which the sample was drawn.

Are humans good or bad at probability?

Humans are terrible at understanding probability. -> The classic example is the coin-flip. If a tossed coin comes up tails 10 times in a row, most people will expect it to come up heads on the next flip. The reality, as we know if we think it through, is that the chance of either heads or tails is the same 50/50.

What is the central limit theorem relationship to probability?

In probability theory, the central limit theorem (CLT) states that the distribution of a sample variable approximates a normal distribution (i.e., a "bell curve") as the sample size becomes larger, assuming that all samples are identical in size, and regardless of the population's actual distribution shape. + The CLT gives us a certain distribution over our estimations. We can utilize this to pose an inquiry about the probability of an estimate that we make.

What is the difference between "Lower-bound" vs "upper-bound"?

Lower bound: a value that is less than or equal to every element of a set of data. Upper bound: a value that is greater than or equal to every element of a set of data.

Population parameters are fixed

Population parameters are fixed

What is measurement error?

The difference between what you measured...and what the real or "true" measure of the variable is. The amount of error in any measure varies. There could be considerable error in one measurement and very little in the next. Measurement error exists in both direct measures (like blood pressure) and indirect measures (like pain).

What do you need to calculate a z-score?

The formula for calculating a z-score is is z = (x-μ)/σ, where x is the raw score, μ is the population mean, and σ is the population standard deviation. As the formula shows, the z-score is simply the raw score minus the population mean, divided by the population standard deviation. -> MEAN, SD, RAW SCORE

Know the tradeoff between confidence and precision

The more precise our estimate, the less certain we are of it; the more confident we are in our estimate, the less precise our "guess" is.

Know what happens to the width of a confidence interval when you: Increase or decrease your sample size...

The width of a confidence interval decreases as the sample size increases and increases as the confidence level increases. Explanation: Larger samples give narrower intervals. We are able to estimate a population proportion more precisely with a larger sample size.

Know what happens to the width of a confidence interval when you: Increase or decrease your standard deviation...

The width of the confidence interval decreases as the sample size increases. The width increases as the standard deviation increases. The width increases as the confidence level increases (0.5 towards 0.99999 - stronger).

What pieces of information do you need to calculate a z-score?

To calculate z-scores, take the raw measurements, subtract the mean, and divide by the standard deviation

What pieces of information do you need to calculate a confidence interval?

To compute a 95% confidence interval, you need three pieces of data: -> The mean (for continuous data) or proportion (for binary data) -> The standard deviation, which describes how dispersed the data is around the average. The sample size.

Multiplication Rule

To determine the probability, we multiply the probability of one event by the probability of another.

Addition Rule of Probability

Used to determine the probability that at least one of two events will occur. P(A or B) = P(A) + P(B) - P(AB)

Know what happens to the width of a confidence interval when you: Increase or decrease your confidence level...

What happens to the width of your confidence interval as you increase your confidence level?: -> The width increases as the standard deviation increases. -> The width increases as the confidence level increases -> The width of the confidence interval decreases as the sample size increases

Z Distribution

a distribution of standardized scores from a sample (symmetrical, unimodal and has an area of 1.00) + Distribution has mean of ZERO and SD of ONE

Sample

a subset of the population

The CLT states that any time descriptive statistics are computed from

an infinite number of large samples, the resulting sampling distribution will be normally distributed.

Problem with population

are usually far too large for researchers to examine directly. e.g; There are millions of male military veterans in the United States, thousands of communities nationwide, and approximately 1.5 million prison inmates. Nobody can possibly study any of these populations in its entirety.

Criminal justice and criminology researchers, however, often use

continuous variables

Probability Theory

grounded in assumptions of infinite trials, and concern what is expected over the long run

What is a z-score?

is a numerical measurement that describes a value's relationship to the mean of a group of values. Z-score is measured in terms of standard deviations from the mean. If a Z-score is 0, it indicates that the data point's score is identical to the mean score.

Sampling Distribution

is a probability distribution of a statistic that is obtained through repeated sampling of a specific population. It describes a range of possible outcomes for a statistic, such as the mean or mode of some variable, of a population.

Why is the Central Limit Theorem important?

is important for statistics because it allows us to safely assume that the sampling distribution of the mean will be normal in most cases. This means that we can take advantage of statistical techniques that assume a normal distribution + it enables us to disregard the shape of the population when the value of n is relatively large. -> understand rarity of events and let us know that the normal distribution will be symmetric (SAMPLE LARGE NUMBER OF CASES)

Parameter

is just like a statistic except that it describes a population (POPULATION = PARAMETER)

The CLT is integral to statistics because

of its guarantee that the sampling distribution will be normal when a sample is large. → The CLT saves the day by ensuring that even skewed or kurtotic variable will produce normal sampling distributions

Unimodal

one peak

Formula of probability

p(A) = the number of times event A can occur / the total number of possible outcomes

What is the gambler's fallacy?

refers to our belief that the probability of a random event occurring in the future is influenced by the past history of that type of event occurring. EXAMPLE: -> Someone in an abusive relationship might believe that, because his/her partner has been abusive for a week, he/she is "due" for a nice streak.

What sets sampling distributions apart from empirical distributions is that

sampling distributions are created not from raw scores but, rather, from sample statistics. → These descriptors can be means, proportions, or any other statistic.

Confidence Interval

statistical range, with a given probability, that takes random error into account

How to Calculate Probability:

step 1: Determine total number of trials step 2: determine the number of trials that be success step 3: divide the number of successes by the number of trials

Benefit of Sampling Distribution (normally distributed) is

that the standard normal curve can be used

What is sampling error?

the difference between the results of random samples taken at the same time. is the difference between a population parameter and a sample statistic used to estimate it. For example, the difference between a population mean and a sample mean is sampling error. Sampling error occurs because a portion, and not the entire population, is surveyed.

Z-score tells you the distance between

the mean and an individual raw score, as well as whether that raw score is GREATER OR LESS than the mean

Standard error:

the standard deviation of a sampling distribution

Population

the whole/entire quantity being studied