t distribution (Student’s t distribution)

In the previous discussions, it was shown that when the population is normally distributed, or when the sample size is large enough, the sampling distribution of the mean is normally distributed. And of course, the bell curve is very handy to use.

However, in many cases where we can only obtain small sizes, the normal distribution does not hold true. Instead, we use the t distribution which is the distribution of t-scores.

t = t scorex = sample mean = population mean = sample standard deviation = sample size

EXAMPLE 1Find the t-score for a

sample size of 16 taken from a population with mean 10 when the sample mean is 12 and the sample standard deviation is 1.5.

Degrees of freedom- the number of

observations in a data set that are free to change without changing the mean.For a single group test𝑑𝑓=𝑁−1

For 2-group tests𝑑𝑓=𝑁1+𝑁 2−2

PROPERTIES of a t distribution1. The distribution has mean 0.2. The distribution is

symmetric about the mean.3. The variance is equal to .4. The variance is always

greater than 1, but approaches 1 when df gets bigger.

What if n approaches infinity?

The t distribution also approaches the standard normal distribution.

Increasing the sample size will….

Increasing the degrees of freedom will…

Increasing the degrees of freedom and sample size will make the t distribution approach a normal distribution.

The critical value is the thin line between rejection and acceptance.

The confidence interval is actually the acceptance region.

The t TABLE

1. The critical region appears at the top.

2. The degrees of freedom are on the leftmost section.

3. Confidence interval are at the bottom.

EXAMPLE 2Find the t-score below

which we can expect 99% of sample means will fall if samples of size 16 are taken from a normally distributed population.

EXAMPLE 2SOLUTION

1−𝛼=0.99𝛼=0.01

𝑑𝑓=𝑛−1

𝑡 0.99=−𝑡 0.01𝑑𝑓=15

𝑡 0.99=−𝑡 0.01=−2.602

EXAMPLE 3If a random sample of size

25 drawn from a normal population gives a mean of 60 and a standard deviation of 4, find the range of t-scores where we can expect to find the middle 95% of all sample means.

EXAMPLE 4Compute the probability

that ).