Confidence Intervals and Significance Testing in the World of T

Welcome to the Real World…

The World of T

T

When Would We Use T?T is used for testing means (averages)…

Looking for Statistically Significant differences

AquacultureCalculating and comparing

Tank Flow Rate to Ideal Flow Rate

HMMM… T – Testing in Research????

What’s T-like?T is a density curve

Symmetric about Zero, single peaked, “bell” shaped

T’s variation depends on sample sizeRemember, samples become less variable as they

get larger

Degrees of FreedomT makes an adjustment for each sample size by

changing the degrees of freedom

Basically gives us a new T to work with for each sample size!!

Let me break this down

really simply for you…

For T- Testing, we’re still testing for

Population Means, but we only need

SAMPLE data!!

Let’s See What Makes

T, T…

Check Me Out!!Wow! A tailor made T for each sample!!

T’s Statistic

__

xt

sn

Standard Error

Sample Standard

Dev.

With n-1 Degrees of Freedom

One Sample T Statistic

This is that personal touch

for each different sample…

Reading the T-Table

P-Value Area to the right of tArea to the left of –t2(P) for two-sided

This is for the T-Table… Let’s Practice

Degrees of Freedom (df)•Left hand column of chart•Different T-Distribution for each sample size•Larger the sample, the closer to Normal the T distribution

T-Statistic•Located in MIDDLE of chart•Leads to the p-value or vice versa

Table Practice Find the t-statistic for the following:

1) 5 dof; p = .05 (right)

2) n = 22; p = .99 (left)

3) 80% CI; n = 18

Find the p-value for the following:

1) 5 dof; t = 3.365

2) n = 12; t = 1.856

3) n = 67; t = 2.056

t = 2.015t = 2.518

t = 1.333

p = .01.025 < p <.05.02 < p <.025

Notice the t-statistic is limited to certain values

on your table!!!

What happens if you get a T that’s not on your table?

Then What?

You will simply say you’re p-value is BETWEEN 2 values!!

Significance Testing for Population Mean (unknown σ)With these tests you are given an alpha

level against which you test your p-value *(Standard level = .05):

p ≤ a – Reject the null; accept the Ha

p > a – Fail to reject the nullHa: µ > µ0 Ha: µ < µ0

Ha: µ ≠ µ0

When to Use T (or not to) *Conditions*

You need to show this check of

conditions after you write your

hypotheses…

Graph your distribution for

samples less than 40 to determine level of

normality!!!

If your HISTOGRAM is skewed, either

scrap the t-test or talk about the

questionability of the results!!

Sample Size Distribution Proof

n <15 Needs to be normal

Histogram or Stem Plot

15 < n < 40 No STRONG outliers or skewness

Histogram or Stem Plot

n < 40 No restrictions

Not Needed

Simple Random Sample

Significance Testing for Population MeanFor T – Tests (testing for population

mean with sample mean and standard deviation…

We use the Same Basic steps as in all Hypothesis

Testing

State the Ho and Ha in symbols and

contextFind the T-StatisticFind the p-value

from the t-statistic w/ n-1 degrees of

freedom

Compare your p-value to the specified , and make your decision in

context

Is My Flow Rate Ideal?Mrs. Luniewski has claimed the ideal flow rate for fish growth is 22 L/min of water flow. You’ve decided to check to see if there’s a statistically significant difference between the ideal flow rate and the flow rate of your tank. You take a SRS of 50 rates from the past week and find your tank has an average flow rate of 14 with a standard deviation of 1.36. At a 5% significance level, is your flow rate significantly less than the ideal flow rate?Ho: µ = 22

L/minHa: µ < 22L/m

14 2241.6

1.36

50

t

t = -2.4072

(n – 1) df = 49 (round down to 40 for table)

P is less than .0005

Since p is less than .0005, which is less than .05, we have

statistically significant evidence that our flow rate is SIGNIFICANTLY less than the

ideal rate. This would help us identify potential problems

with the tank…

Matched Pairs T-TestingMatched Pairs Test

Used when taking same measurements on same media over different time period

“Difference” between values is THE data

Ho = µdiff = 0 [µdiff = (µ1 - µ2)]

Ha = µdiff < or > or ≠

Flow Rate of Tank Pre-treatment

Flow Rate of Tank Post-treatment

Difference(Post – Pre)

12 9.5 2.5

11 8 3

9 5 4

You use this difference column to get your Sample Mean and Sample Standard

Deviation…

Matching Flow Rates…Now you have applied a water treatment to the

tanks, hoping to make a difference in the average flow rate of your tank. To check to see if the difference is statistically significant, you collect 3 measurements (not really enough!!) from the tanks at the exact same times they were collected pre-treatment. Test to see if there is a significant difference post-treatment.

Because the measurements were taken on the same tank, at the

same time, this would be considered a matched pairs test.

You would need to adjust your hypotheses accordingly and use

the difference between the data as your data source.

Ho: µdiff = 0Ha: µdiff < 0 (Increase in flow rate)

Matching Flow Rates (cont)Flow Rate of

Tank Pre-treatment

Flow Rate of Tank Post-treatment

Difference(Post – Pre)

12 9.5 2.5

11 8 3

9 5 4

Sample Average Difference = 3.17Sample Std Dev = .7638

3.17 07.19

.7638

3

DF = 2 .005 < p < .01

REJECT Ho and conclude the treatment made a significant

increase in flow rate…