confidence intervals and significance testing in the world of t welcome to the real world… the...
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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…