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  • We use Z-scores to describe individuals according to their score and their standard deviation
  • Quite simply a persons Z-score is how many standard deviations there score is above or below the mean
  • Ex. If a for a population of people the mean for their number of hours slept per day is 7 and the standard deviation is 1
  • And if you as a member of that population sleep 9 hours per day
  • Then we can say that you sleep 2 standard deviations more than the mean of the population
  • Thus you will have a z-score of 2
  • If you sleep 6 hours per day then you sleep 2 standard deviations less than the mean
  • And thus your z-score will be -2

i.e. People who have scores higher than the mean will have positive z-scores and people who have scores lower than the mean will have negative z-scores

  • Step 1: For any hypothesis test we start by assuming that the null hypothesis is true and then we measure whether the result we have is strange given our assumption
  • Step 2: We determine the range of results that we will consider strange

For this course any result that only has a 5% chance of occurring or less will be considered a strange result

  • Step 3: If we get a strange result is then we say that the evidence supports the test hypothesis

If we don't then we say that we don't have enough evidence to say that the null hypothesis is false (More formally we say that we fail to reject the null hypothesis

  • Ex. We have a sample of people suffering from sleep deprivation and we know the mean (6) and standard deviation (1) of the stress levels of the entire population of people suffering from sleep deprivation.
  • We want to use this sample for an experiment measuring the effects of sleep deprivation on depression.
  • Before we do so we suspect that one of the people in our sample (Nadine) is not sleep deprived and has snuck into our sample
  • We will use hypothesis testing, Nadine's stress levels, and the population's stress levels to examine whether she belongs to this population (of sleep deprived individuals) or not.

P.S. Notice how we said our results "support" the test hypothesis. We did not say "prove". This is because we can never be 100% certain, and thus in psychology we never use the word prove

  • We cannot usually measure information about the entire population
  • Instead we can try and pick a small group of people that can accurately represent the population we want to make predictions for
  • This is called sampling
  • The best method is called random sampling (randomly picking members of your group from a list of everyone in this population)
  • This is different from haphazard sampling (which occurs frequently in psychology) where you call for participants and accept whomever applies for it
  • It's best to find a compromise between these two methods when you can
  • And when you cannot always keep in mind how your method of sampling will affect how you will make sense of your data and results

Normal Distribution

Z-Scores

  • A normal distribution is a uni-modal and symmetric distribution
  • Psychologists like to assume that all the variables they measure follow a uni-modal distribution
  • So long as the true distribution isn't extremely different than the normal distribution we can safely make this assumption

Hypothesis Testing

  • The assumptions or predictions we want to make when using inferential statistics are called hypothesis
  • For each assumption/prediction we have a Null and a Test hypothesis
  • The test hypothesis is the existence of the prediction that you are making
  • The null hypothesis is its absence
  • i.e. prediction: The amount of sleep you get per day affects your stress levels
  • Test Hypothesis (H1): Stress levels change according the the number of hours you sleep
  • Null Hypothesis (Ho): The number of hours you sleep per day have no effect on your stress levels

P.S. The null hypothesis is not always the opposite of the test hypothesis, but it is always the absence of it

Normal Dist. Cont.

Test Hypothesis

  • There are 2 types of test hypothesis
  • 1-tailed hypothesis: When the effect you are looking for is either specifically positive or negative (ex. Stress increases depression or stress reduces depression)
  • 2-tailed hypothesis: When you don't know whether the effect is negative or positive, only that it exists (ex. Stress changes people's level of depression)
  • You must always decide whether you will be using a 1-tailed or 2-tailed test hypothesis BEFORE you analyze your data
  • Since the normal distribution always follows the same shape then we can use the standard deviation to figure out the percentage of people that have scored more or less than a specific score

Hypothesis Testing

The following text contains an example of of how all these principles are integrated and form hypothesis testing in psychology. It will not be a test that will occur in a real world situation, but it will serve to prime your for the types of tests we will be using in the coming classes.

Probability

Example

The p value (or probability) is the ratio of an outcome occurring compared to all other possible outcomes

Steps

Ex. continued

Stress levels for the population:

Mean= 6, Standard deviation=1

Nadine's stress level = 8.5

Step 1: Assume that Nadine belongs to the population

Step 2: If Nadine belongs to the population it would be strange if her z-score was greater or less than 2

This is because in that case only 5% of the population would have a score similar to hers

Step 3: Calculate Nadine's z-score

Nadine's Z-score = (8.5 - 6)/1 = 2.5

Nadine's z-score is greater than 2 and thus if Nadine really is part of this population then this would be a really strange result

  • Given what we know we can say that our results support the assumption that Nadine isn't part of our sleep deprived population

This p-value can be expressed as a fraction, a percentage, or using decimals

Sampling

Principles for inferential statistics & Hypothesis Testing

  • Ex. If for a population of depressed people the mean depression score is 8 and the standard deviation is 1
  • Then anyone with a depression score of 10 is 2 standard deviations above the mean
  • And anyone with a depression score of 6 is 2 standard deviations below the mean
  • Using this infomation we can determine that 2.5% of our population will have a depression score of 10 or above
  • We can say the same for anyone with a depression score of 6 or below
  • We can Also say that 95% of our population will have a depression score that lies between 6 and 8

P.S. These number are just approximations, we will use the accurate numbers later in this course.

  • This is the equation used for calculating a person's z-score
  • X is the person's raw score
  • M is the mean of the population
  • SD is the standard deviation of the population
  • You will not have to memorize this equation or any other equation in this course
  • Ex. For a dice the p-value or probability of getting a 6 when rolling a dice once is 1/6 or 16.7% or 0.167
  • The probability of getting a number that's 2 or lower after rolling the dice once is 2/6 or 33.3% or 0.333
  • For a coin the probability of it landing on heads when flipping the coin once is 1/2 or 50% or 0.5
  • Using this same logic we can say that for any normally distributed population the probability of me running into someone with a z-score between 2 & -2 (for any variable I'm measuring, ex. depression) is 95%
  • We can also say that the probability that I would run into anyone with a different score is only 5%
  • So I only have a 5% chance of running into someone with a z-score that's greater than 2 or smaller than -2
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