PSY 138 Exam 2 SG

The Unit Normal Table: Example: What are the proportions below and above z = +0.24 & z = -0.24?

For z = +0.24 1.Column B indicates proportion below 0.5948 2.Column C indicates proportion above 0.4052 For z = -0.24 1. Column C indicates proportion below 0.4052 2. Column B indicates proportion above 0.5948

Sampling error

the amount of error between a sample statistic and its corresponding population parameter

In a population with a mean of µ = 50, a score of X = 42 corresponds to z= -2.00. What is the standard deviation for the population?

**Need to rearrange z-score formula σ = (X-µ)/z So, σ = (42-50)/-2 = -8/-2 = 4

Properties of a standardized distribution:

1) Shape will not change 2) Mean will always be 0 3) Standard deviation will always be 1

A distribution of sample means is a distribution of sample ______1_______ not _______2______.

1) Statistics (means) 2) Scores

Two conditions for random samples:

1. Each individual in a population has an equal chance of being selected 2. Probabilities must stay constant from one selection to the next (sampling with replacement)

Constructing a Sampling distribution

1. Select a random sample of a specific size (n) from a population, compute the mean, place it in a frequency distribution 2. Select another random sample with the same n, compute the mean, add it to distribution 3. Continue until you have a complete set of all possible random samples

Important about Probability:

1. always positive 2. can only be between 0 and 1 (0% and 100%)

What does a z score tell us?

1.To find the exact location of the score within a distribution 2. To allow comparisons across distributions

Example of Random Sampling: p (any card in a deck of cards) = 1/52 Draw a random sample of cards Draw 1: you draw jack of diamonds, you don't put the card back in Draw 2: what is the probability?

1/51

Random sampling

A method of selecting a sample from a statistical population in such a way that every possible sample that could be selected has a predetermined probability of being selected

Location of Z-Scores: Z= -2

At the far left (extreme left tail)

Location of Z-Scores: Z=+2

At the far right (extreme right tail)

The Unit Normal Table

Body=larger part Tail=smaller section For a positive z score: 1.Column B indicates proportion below 2.Column C indicates proportion above For a negative z score: Switch B and C 1.Column C indicates proportion below 2.Column B indicates proportion above

Location of Z-Scores: Z = 0

In the center (at the mean)

Distribution of sample means (sampling distribution of the mean)

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

What do we need to determine probabilities or proportions for a normal distribution?

Need X (raw score) Need z for X (to calculate z we need mean and std. deviation) Use the unit normal table

standard normal distribution table

Percents between numbers: 0 to -1 or 0 to +1: 34.13% -1 to -2 or +1 to +2: 13.59% Beyond -2 or +2: 2.28%

Probability

Possible outcomes A, B, C, D etc. Probability of A = (𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑐𝑙𝑎𝑠𝑠𝑖𝑓𝑖𝑒𝑑 𝑎𝑠 𝐴)/(𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠)

Probability Example: Complete deck of cards: 52 Probability of selecting an ace is: ?

Probability of A = (𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑐𝑙𝑎𝑠𝑠𝑖𝑓𝑖𝑒𝑑 𝑎𝑠 𝐴)/(𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠) Complete deck of cards: 52 Probability of selecting an ace is: 4/52 = 1/13

The ______ of the z-score tells us if the score is above or below the mean

Sign Positive (+): above the mean Negative (-): below the mean

The Unit Normal Table: Example: The population parameters for the SAT are: μ = 500, σ = 100, and it is Normally distributed Suppose that you got a 630 on the SAT. What percent of the people who take the SAT get your score or worse?

So, z = (630-500)/100 = 1.3 So a z(1.3) =.0968 tail. 9.68% of scores are above z (1.3) or 630. 100% - 9.68% = 90.32% are below

Standardizing a distribution with predetermined mean and SD: Example M=42 SD=14 S Std. distribution M=100, SD=10, X=56

Step 1: Calculate the z score for your score in the original distribution (M=42, SD=14) using X=56. Z = (56-42)/14 = 14/14 = +1.00 Step 2: Calculate the raw score corresponding to the z score you calculated but using M and SD info from the new distribution (M=100, SD=10), and Z= +1.00. X = (Z x SD) + M = (1 x 10) + 100 = 10 + 100 = 110 So, 56 becomes 110 in the new distribution

In terms of above/below and number of standard deviations, what does a Z-score = +2 tell us?

The score is above the mean by two standard deviations.

The number value of the z-score tells us the distance between each score and the mean in terms of number of _______________ __________________.

standard deviations

What does it mean to standardize a distribution?

Transforming every score in a distribution into a z-score

True/False Probability can be expressed as fractions, decimals, or percentages.

True; For Example Tossing a coin: heads and tails p(heads) = 1/2 = 0.5 = 50%

True or False: Z-Scores can be used as both inferential and descriptive statistics?

True; The fact that z-scores identify exact locations within a distribution means that z-scores can be used as descriptive statistics and as inferential statistics.

Why is standardizing a distribution helpful?

You can compare scores across distributions -We can only do this with standardized distributions-

Given: X=5 M=3 SD=2 What does z equal?

Z = (X-M)/SD So, Z = (5-3)/2 = 2/2 = +1.00

How do Z-scores function as descriptive statistics?

Z-scores describe exactly where each individual is located.

How do Z-scores function as inferential statistics?

Z-scores determine whether a specific sample is representative of its population, or is extreme and unrepresentative.