Statistics 2

fact

Although the z-score values change signs (+ and −) from one side to the other, the proportions are always positive.

A population with μ and σ is transformed into z-scores. After the transformation, what is the mean for the population of z-scores?

0

fact

A z-score near 0 indicates that the score is close to the population mean and therefore is representative. A z-score beyond +2.00 (or −2.00) indicates that the score is extreme and is noticeably different from the other scores in the distribution.

fact

Because the normal distribution is symmetrical, the proportions on the right-hand side are exactly the same as the corresponding proportions on the left-hand side

What location in a distribution corresponds to ?

Below the mean by a distance equal to 2 standard deviations

What location in a distribution corresponds to z=-2.00 ?

Below the mean by a distance equal to 2 standard deviations

the process of transforming X values into z-scores

Each z-score tells the exact location of the original X value within the distribution. The z-scores form a standardized distribution that can be directly compared to other distributions that also have been transformed into z-scores.

Central limit theorem:

For any population with mean μ and standard deviation σ , the distribution of sample means for sample size n will have a mean of μ and a standard deviation of σ and will approach a normal distribution as n approaches infinity.

fact

In inferential statistics, z-scores provide an objective method for determining how well a specific score represents its population

Random sampling requires sampling with replacement. What is the goal of sampling with replacement?

It ensures that the probabilities stay constant from one selection to the next.

fact

The body always corresponds to the larger part of the distribution whether it is on the right-hand side or the left-hand side. Similarly, the tail is always the smaller section whether it is on the right or the left

fact

The distribution of z-scores will always have a standard deviation of 1.

Shape

The distribution of z-scores will have exactly the same shape as the original distribution of scores. If the original distribution is negatively skewed, for example, then the z-score distribution will also be negatively skewed. If the original distribution is normal, the distribution of z-scores will also be normal.

expected value of M

The mean of the distribution of sample means is equal to the mean of the population of scores, μ ,

Explain how a z-score identifies an exact location in a distribution with a single number.

The sign of the z-score tells whether the location is above (+) or below (-) the mean, and the magnitude tells the distance from the mean in terms of the number of standard deviations.

fact

The z-score distribution will always have a mean of zero

fact

Transforming raw scores into z-scores does not change anyone's position in the distribution

Under what circumstances would a score that is 15 points above the mean be considered an extreme score?

When the standard deviation is much smaller than 15

percentile rank

for a specific score is defined as the percentage of the individuals in the distribution who have scores that are less than or equal to the specific score

standard error of M

he standard deviation of the distribution of sample means, σ , The standard error provides a measure of how much distance is expected on average between a sample mean (M) and the population mean μ .

sampling distribution

is a distribution of statistics obtained by selecting all the possible samples of a specific size from a population.

standardized distribution

is composed of scores that have been transformed to create predetermined values for μ and σ . Standardized distributions are used to make dissimilar distributions comparable.

distribution of sample means

is the collection of sample means for all the possible random samples of a particular size (n) that can be obtained from a population.

sampling error

is the natural discrepancy, or amount of error, between a sample statistic and its corresponding population parameter.

The purpose of z-scores, or standard scores

is to identify and describe the exact location of each score in a distribution.

unit normal table

lists relationships between z-score locations and proportions in a normal distribution.

Independent random sampling

requires that each individual has an equal chance of being selected and that the probability of being selected stays constant from one selection to the next if more than one individual is selected.A sample obtained with this technique is called an independent random sample or simply a random sample.

random sampling

requires that each individual in the population has an equal chance of being selected.

z-score

specifies the precise location of each X value within a distribution.The sign of the z-score (+ or −) signifies whether the score is above the mean (positive) or below the mean (negative).The numerical value of the z-score specifies the distance from the mean by counting the number of standard deviations between X and μ .

law of large numbers

states that the larger the sample size (n), the more probable it is that the sample mean will be close to the population mean.

fact

the distribution for the sample of z-scores will have the same shape as the original sample of scores.

fact

the sample of z-scores will have a mean ofM2=o .

fact

the sample of z-scores will have a standard deviation of .