Chapter 9 - Single-Sample t-test

Question 1

For Single Sample t-tests, there is no population standard deviation (sigma).

PURPOSE: used to test whether a sample mena is similar or different to a population mean with an unknown population standard deviation.

We want to solve a problem: Q = do older adults score higher than the regular population on the UCLA Loneliness Scale?

KNOWN: N= 62 older adults & M = 47.12 (mean of sample)

BUT! I don’t have the population standard deviation (sigma). I can’t standardize the data. So estimate it from the sample’s deviation?

Answer 1

SO WHAT TO DO…because the sample estimate will still give me a smaller value than my population value.

Make 2 corrections:

1. one to improve our estimate : change the denominator of the below formula AND changed the letters (SD –> s). This little ‘s’ means we are telling others that we are using a ‘sample’ to estimate the standard deviation. And the N-1 means we are making the estimate value a little higher by removing one of the participants in the denominator.

Question 2

Also, we can no longer use the z-distribution b/c we don’t have the pop. standard deviation. If we would underesimate the devation, that would make the z-score too high.

**So the T-distributions is used instead.
**
Like the z-distribution, the mean t-score = 0
They is also more variability because they are flatter and spread out. They are many versions of the t-distributions. They are more conservative than z-distribution.

(so the t-statistic is MORE EXTREME than the z-statistic, meaning it is a MORE CONSERVATIVE test.

Answer 2

the # of participants (N) don’t necessarily change the distribution, but rather the degrees of freedom (all the values can vary but the last value has to be a specific value in order to work out to the given mean)

Question 3

Past 120 degrees of freedom means you should always choose the lesser # in order to be more conservative.

Answer 3

Will be using a DISTRIBUTION OF MEANS – this means the Central Limit Theorem to determine the MEAN & STANDARD ERROR and basically reverse engineer everything to find the populations standard deviation.

USING
1. MEAN of the population
2. STANDARD DEVIATION of the sample

Question 4

1st STEP:

Answer 4

2nd STEP:

Question 5

3rd STEP:

use the Central Limit Theorem to determine the mean & standard error of the comparison distribution.

Mean of distribution µ = mean of population µM

The sM (standard error) = we are using an estimate of the population standard deviation to determine our standard error.

Answer 5

4th STEP:

Set up a rejection level & critical values

the graph is set up as a 1-Tail ? WHY? because looking to see if the OLDER adults score higher loneliness.

go to B-2 for t-distributions, using the df value

Question 6

Step 5: Calculate the t-statistic

Answer 6

Step 6:

Question 7

C)

Answer 7

Question 8

Confidence Intervals : allow us to say that we think a MEAN is between this lower value and this higher value within a population, with a certain level of confidence, using a mean. It’s a range of plausible values.

1. Start with the 2.5% in each tail (2-tailed)
2. use B-2 table, look under the right side at the 2-tail part, look under 0.05 (alpha), AND match it up with the degrees of freedom (60 in this case)
3. t-values are -2.001 and +2.001

Answer 8

4. Transform t-values into raw scores.
5. Figure out sM (standard error)
6. use those 2 formulas to find the lower and upper ranges of the confidence interval
7. 7.

Question 9

Not only do we have the range of plausible values, we can use this range for** Hypothesis Testing**.

Answer 9

Cohen’s d formula

Watch out for the little ‘s’ in the denominator!

0.206 = a small effect

In order to increase the effect size though, need more participants!

t(61) =1.62, p> 0.05, d= 0.206

Question 10

CONCLUSION

Answer 10

STATISTICAL POWER = just means the power that we’ll find real results