Psy 10B

A population has u = 50 and o = 10. If these scores are transformed into z-scores, the population of z-scores will have a mean of ___ and a standard deviation of ____.

0 and 1

What proportion of a normal distribution is located in the tail beyond a z-score of z = -1.50?

0.0668 **Why is this? (6.2)

Characteristics of the Distribution of Sample Means

1. Sample means should pile up around the population mean. Samples are representative of the population. 2. The pile of sample means should tend to form a normal-shaped distribution. Logically, most of the samples should have means close to the "u." 3. The larger the sample size, the closer the sample means should be to the population mean.

Using z-scores to standardize a distribution

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. Because each individual score stays in its same position within the distribution, the overall shape of the distribution does not change. 2. The Mean: The z-score distribution will always have a mean of zero. 3. The Standard Deviation: The distribution of z-scores will always have a standard deviation of 1. The advantage of having a standard deviation of 1 is that the numerical value of a z-score is exactly the same as the number of standard deviations from the mean.

Unit Normal Table

1. 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. 2. 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. The table does not list negative z-score values. To find proportions for negative z-scores, you must look up the corresponding proportions for the positive value of z. 3. Although the z-score value changes signs (+ and -) from one side to the other, the proportions are always positive. Thus, column C in the table always lists the proportion in the tail whether it is the right-hand tail or the left-hand tail.

Standardized Distributions to create new values for u and o is a two-step process

1. The original raw scores are transformed into z-scores. 2. The z-scores are then transformed into new X values so that the specific u and o are attained.

The Shape of the Distribution of Sample Means

1. The population from which the samples are selected in a normal distribution 2. The number of scores (n) in each sample is relatively large, around 30 or more.

Standardizing a Sample Distribution

1. The sample of z-scores will have the same shape as the original sample of scores. 2. The sample of z-scores will have a mean of Mz=0. 3. The sample of z-scores will have a standard deviation of Sz=1

Computing z-scores for samples

1. The sign of the z-score indicates whether the X value is above (+) or below (-) the sample mean 2. The numerical value of the z-score identifies the distance from the sample mean by measuring the number of sample standard deviations between the score (X) and the sample Mean (M.)

A population with u = 85 and o = 12 is transformed into z-scores. After the transformation, the population of z-scores will have a standard deviation of____

1.00 ***Why is it? (5.4)

A distribution with u = 47 and o = 6 is being standardized so that the new mean and standard deviation will be u = 100 and o = 20. What is the standardized score for a person with X = 56 in the original distribution?

130

A vertical line is drawn through a normal distribution at z = -1.00. How much of the distribution is located between the line and the mean?

34.13% ***How do you get this? (6.2)

If a sample of n =25 is selected from a normal population with u = 80 and o = 10, then what sample means form the boundaries the separate the middle 95% of all sample means from the extreme 5% in the tails?

76.08 and 83.92 ***

Simple random sample

A sample obtained by this process is called a SRS

Sampling Error

A sample typically will not provide a perfectly accurate representation of its population. More specifically, there typically is some discrepancy (or error) between a statistic computed for a sample and the corresponding parameter for the population. In general there will be some discrepancy, or sampling error, between the mean for a sample and the mean for the population from which the sample was obtained.

Sampling dsitribution

Distribution of statistics obtained by selecting all the possible samples of a specific size from a population

If a sample is selected from a normal population with u = 50 and o = 20, which of the following samples is extreme and very unlikely to be obtained?

M = 45 for a sample of n = 100 scores ***

Standard Error

Sample means are relatively close to the population mean. These samples provide a fairly accurate representation of the population. On the other hand, some samples produce means that are out in the tails of the distribution, relatively far from the population mean. These extreme sample means do not accurately represent the population. The standard error provides a way to measure the "average," or standard, distance between a sample mean and the population mean.

Central Limit Theorem

The theory that, as sample size increases, the distribution of sample means of size n, randomly selected, approaches a normal distribution. It describes the distribution of sample means by identifying the three basic characteristics that describe any distribution: shape, central tendency, and variability.

Under what circumstances would a score that is 15 points above the mean be considered to be near the center of the distribution?

When the population standard deviation is much larger than 15 *** explain (5.7)

Probability

for any specific outcome is defined as a fraction or a proportion of all the possible outcomes. If the possible outcomes are identified as A,B, C, D, and so on, then Probability of A = (number of outcomes classified as A) / (total number of possible outcomes)

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

A random sample is obtained from a population with u = 80 and 0 =10 and a treatment is administered to the sample. Which of the following outcomes would be considered noticeably different from a typical sample that did not receive the treatment?

n = 100 with M = 83 ***

For samples selected from a population with u=40 and o=20, what sample size is necessary to make the standard distance between the sample mean and the population mean equal to 2 points?

n=100

For a particular population, the standard distance between a sample mean and the population mean is 5 points for samples of n=4 scores. What would the standard distance be for samples of n=16 scores?

n=2.5 points *****

An independent random sample

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.

Random sample

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

Law of Large Numbers

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

The distribution of Sample Means

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

Expected Value of M

the mean of the distribution of sample means is equal to the mean of the population of scores

Sampling Error

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

Standard Error of M

the standard deviation of the distribution of sample means; it provides a measure of how much distance is expected on average between a sample mean (M) and the population mean.

Under what circumstances would a score that is 20 points above the mean be considered to be an extreme, unrepresentative value?

when the population standard deviation is much smaller than 20. (5.7)

What z-score separates the lowest 10% of the distribution from the rest>

z = -1.28 *** I don't understand this (6.2)

Z-score Formula

z=x-u/o