Statistics: Understanding Z-scores, T-scores, and T-tests, Study notes of Descriptive statistics

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Z-scores, the normal curve, the normal table

T-scores and the t-table

T-tests

T-tests in SPSS

Refresher

Definition of p-value :

The probability of getting evidence as strong as

you did assuming that the null hypothesis is true.

A smaller p-value means that it’s less likely you would get a sample like this if the null hypothesis were true.

A smaller alpha means less of a chance of falsely rejecting the

null. (Also called a Type I error )

A smaller alpha means we want to be more certain about something before rejecting the null. If the p-value is smaller than the alpha, we reject the null hypothesis. (Enough evidence to reject) If the p-value is larger than the null, we fail to reject the null hypothesis. (Not yet enough evidence to reject)

Remember this curve? This is the normal curve.

μ, pronounced ‘mu’ is the mean for normals

σ, pronounced ‘sigma’ is the standard deviation for normal

μ + 2σ refers to the point two standard devs above the mean

2/3 and 95% are proportions, or ratios between a part of a group and that group as a whole. Proportions are useful because they also imply probability. If 2/3 of the data is within 1sd, then if I pick a point at random from that distribution… … there is a 2/3 chance that it will be within 1 standard deviation.

Example: Reading scores Grade 5s reading scores are normally distributed with mean 120 and standard deviation 25. Pick a grade 5 student at random… You have a 95% chance of getting one with a reading score between 70 and 170.

Rather than describe things in terms of 'standard deviations above/below the mean', we shorten this to z-scores.

There's also the standard error, which describes the uncertainty about some measure, such as a sample mean.

The standard error is a measure of the typical amount that that a sample mean will be off from the true mean. We get the standard error from the standard deviation of the data…

• The more we sample from a distribution, the more we know about its mean.

- Standard error decreases as sample size n increases.

Like standard deviation, the calculation of standard error isn’t as important as getting a sense of how it behaves.

- Bigger when standard deviation is bigger.

- Smaller when sample size is bigger.

Tophat: In order to make a standard error two

times smaller (1/5 as large), how many times

as large does the sample need to be?

The z-score can be used to describe how many standard ERRORS above or below a population mean that a sample mean is. (recall sample mean vs. Population mean)