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# Chapter 7 stats.docx

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Western University

Psychology

Psychology 2135A/B

Sandra Hessels

Fall

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Chapter 7 – Hypothesis Testing with z
Tests
THE Z TABLE
With the z test, as with all other hypothesis tests, there are three different ways to identify the exact
same point beneath the normal curve: raw score, z score and percentile ranking.
Z table is the tool that allows us to transition from one to another of these ways to identify a point
Raw Scores, z Scores and Percentages
Allows us to translate the standardized z distribution into percentages and individual z scores or z
statistics into percentile rankings
We learned that
1) about 68% of scores fall within one z score of the mean
2) about 96% of scores fall within two z scores of the mean
3) Nearly all scores fall within three z scores of the mean
Determine the percentage associated with a given z statistic by following two steps:
1. Convert a raw score into a z score
2. Look up a given z score on the z table to find the percentage of scores between the mean and that z
score.
Positive z score is identical to calculating the percentage between the mean and the negative version of
that z score.
EXAMPLE (with positive number)
Step 1: convert her raw score to a z score, as we learned how to do in Chapter 6. (0.98)
Step 2: Look up 0.98 on the z table to find the associated percentage between the mean and Jessica’s z
score (33.65%)
1. Jessica’s percentile rank, the percentage of scores below her score
Add the percentage between the mean and the positive z score to 50%.
Jessica’s percentile is 50% + 33.65% = 83.65%
because it is above the mean, we know that the answer is higher than 50%
2. The percentage of scores above Jessica’s score
Subtract the percentage between the mean and the positive z score from 50%
50% - 33.65% = 16.35%
percentage would be smaller than 50% because the z score is positive.
Alternate: subtract Jessica’s percentile rank of 83.35% from 100% 3. The scores at least as extreme as Jessica’s z score, in both directions
Double 16.35% to find the total percentage of heights that are as far or farther from the mean than is
Jessica’s height
16.35% + 16.35% = 32.70%
EXAMPLE (with negative number)
1. Manuel’s percentile score, the percentage of scores below his score
50% - 46.56% = 3.44%
2. The percentage of scores above Manuel’s score
50% + 46.56% = 96.56%
3. The scores at least as extreme as Manuel’s z score, in both directions
3.44% + 3.44% = 6.88%
rd
EXAMPLE (scored at the 63 percentile, what was her raw score?)
Add a line at the point where approx. 63% of scores fall. – know that this score is above the mean
because 50% of scores fall below the mean and 63% is larger than 50%
63% - 50% = 13%
Look up percentage closest to 13% in the z table. Find associated z score.
Convert that score to a raw score
X = z(sigma) + mew
The Z Table and Distributions of Means
In hypothesis testing, we use means rather than scores because we would always study a sample rather
than an individual.
Z table can also be used to determine percentages and z statistics for distributions of means calculated
from having many people
Same as before but additional step of first having to calculate the mean and the standard error for the
distribution of means
EXAMPLE:
Mean – 554
SD – 99
Sample – 90
Sample mean – 568 Want to know how much better (or worse) students in our department are doing by comparison to the
mean score of the population, z stats make that comparison possible:
Distribution of means has same mean as the distribution of scores for the population, have to find the
standard error of this distribution of means
Then convert to z statistic using what we found above = 1.34
Shade the area in which we are interested, everythin

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