T1WB Normal Distribution and Z-Scores

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Normal Distribution & Hypothesis Testing

• A normally distributed set of scores can be transformed into a standard normal distribution (z-distribution).

• This is done by rescaling the scores to have a mean of 0 (µ = 0) and a standard deviation of 1 (s = 1).

• A z-score represents the number of standard deviations a score lies above or below the mean.

• The z-distribution can determine the probability of a score randomly drawn from the population lying beyond a particular value.

• Formula for z-score: z = (x - μ) / σ

Comparing a Single Score to a Population

• Knowledge of the standard normal distribution can be used to determine if a person's score is significantly different from the population mean.
• The null hypothesis states that any difference between the sample mean and the population mean is due to sampling error.
• It assumes no difference or effect, meaning an individual's score is not significantly different from the population mean.
• The alternative hypothesis suggests that any difference between the sample mean and the population mean is not due to sampling error.
• It states there is a difference or effect (e.g., the individual’s score is significantly different from the population mean).

Testing Hypotheses About Means

• The sampling distribution of the mean is considered

• σ (or σx) represents the standard deviation of scores in the population.

• σx represents the standard error of the means (SEM), which is the standard deviation of the means of samples of a particular size drawn randomly from that population.

• The standard error of the mean measures how much the means of samples of a particular size are expected to vary around the true mean for the population.

• This depends on: the amount of variation in the population of scores themselves (s), where more variation leads to more variability in sample means, and the sample size (N), where a larger sample size leads to less expected difference.

• Given a population of scores with a mean (μ) and a standard deviation (s), the sampling distribution of the mean will have a mean μx = μ.

• Standard deviation: σx = σ / √N (the standard error of the mean).

• The standard normal distribution (z-distribution) can determine the probability that a sample of size N drawn randomly from the population would have a mean at least as extreme as the sample mean.

t Statistic

• A single-sample t-test can compare the mean of a single sample against a population when the population mean is known, but the population variance s² (or standard deviation s) is not known and must be estimated from the sample.

• Formula:

• z = (X - μ) / σx = (X - μ) / (σ / √N)

• t = Observed Difference / Expected Difference = (X - μ) / Sx = (X - μ) / (s / √N)

• The t statistic compares the observed difference between the sample mean and population mean to the average difference expected in samples of that size due to sampling error.

• Unless the sample size is large, the sampling distribution of t does not conform exactly to the z-distribution; it follows Student's t-distribution, which varies with the degrees of freedom (df) of the test.

• T-tables provide critical values of t for different df.

Assumptions of the t-test

• The t-test, like the z-test, is a parametric test based on the following assumptions:
• Random sampling: Scores are randomly sampled from the relevant population.
• Normal distribution: The sampling distribution of the mean should be normal.
• The central limit theorem states that the sampling distribution of the mean will be normal if the population of raw scores is normally distributed or if N is at least 30 .