# t-distribution

using T instead of z

If we use estimated standard error that is based on sample standard deviation (s) then.... /We need to use the t-distribution instead of z-distribution to find area under the normal curve /z uses population standard deviation () while t uses sample standard deviation (s)

T distribution

There is whole 'family' of t distributions. To decide which one to use, need to know two things: 1. Decide what alpha level you will be using 2. Calculate degrees of freedom (df = n-1)

step 3:calculate degrees of freedom

Indicates number of scores in a given sample that are independent & free to vary (sample mean places restriction on one score)

step 1: decide if using T or Z distribution

Ask yourself, "Do I know the population variance or the sample variance?" / In this example we know SAMPLE standard deviation / Therefore we need to use t distribution & calculate t-score

compute test statistic

First calculate standard error Then calculate t-score

what is alpha again?

It is the probability of rejecting null hypothesis when, in fact, it is true

What is alpha again?

It is the probability of rejecting the null hypothesis when, in fact, it is true (type 1 error)

step 4: find corresponding t-statistic

Look up t-statistic in Table B.2 (p. 729) / t-statistic for 0.05 alpha level & 249 df equals 1.96 / In this table, sample size of 249 falls into infinity category

standard error

Remember, standard error is standard deviation of distribution of sample means

find critical value

Remember, we are using t distribution NOT z distribution / Why??? Because we do not know population standard deviation

find critical value

Remember, we are using t distribution NOT z distribution \ Why??? Because we do not know population standard deviation / Are we conducting one-tailed or two-tailed test? One-tailed!

why do we subtract 1 again?

Sample variance (s2) & SD (s) are smaller when computed using sample data as opposed to population data / Samples underestimate amount of variability / Need to inflate sample variance & SD slightly to produce more accurate estimates of population parameters / Therefore, divide SS by n-1 instead of n. This correction factor of n-1 is called DEGREES OF FREEDOM!

using T instead of z

The larger the sample size, the closer tdistribution represents normal curve, & better t-score approximates z-score WHY??? Because larger random samples contain more representative information about population!!!

what if we don't know population sd

Until now, assuming that we know actual variance in population / In reality, this is extremely unlikely. So, what do we do??? /Use ESTIMATE of standard error (slightly different formula) / Also, use t distribution (t-score) instead of z distribution (z-score)

step 6 interpret your findings

We are 95% confident that fathers in the US spend, on average, between 32.91 & 34.10 hours per week with their preschool aged children

step 2:calculate alpha level

We know from original question that we want to use 95% confidence interval so... To calculate alpha, subtract confidence level from 1 (1-.95)

Does alpha level mean the same thing for t-test compared to z-test?

YES!!!

step 5

calculate confidence levels