t-tests mini lecture

Summary

This lesson explains the concept and application of the t-test, a statistical method used when the population variance is unknown and must be estimated from the sample. It clarifies the process of adjusting sample variance to estimate the population variance using degrees of freedom (n-1), and the use of the t-distribution for hypothesis testing. The video also discusses the comparison of sample means to a population mean, the importance of selecting the correct t-distribution based on degrees of freedom, and the robustness of the t-test under normality assumptions. The summary concludes with the steps for calculating the t-score and making decisions about the null hypothesis.

Takeaways

• 📚 In real-world scenarios, we often lack population characteristics like mean, variance, and standard deviation, and instead rely on sample data to estimate these values.
• 🧐 The sample variance is typically lower than the population variance, which can lead to estimation errors if not properly adjusted.
• 🔍 To estimate the population variance, we use a modified formula that divides the sum of squares by the degrees of freedom (n - 1), where n is the sample size.
• 📉 The degrees of freedom represent the number of independent values in the sample that can vary before the last value can be determined.
• 📊 When comparing sample means to a population, we use the T-distribution if the population variance is unknown, as it accounts for the increased variability in sample estimates.
• 📈 The T-distribution is flatter and has more extreme values compared to the normal distribution, with different distributions corresponding to different degrees of freedom.
• 📝 The comparison distribution's cutoff scores are determined by the degrees of freedom and the type of hypothesis (one-tailed or two-tailed), with tables provided for reference.
• 🔢 The variance of the distribution of means is calculated by dividing the estimated population variance by the sample size (n).
• 📐 The standard deviation of the distribution of means, or the standard error, is found by taking the square root of the variance of the distribution of means.
• 🔄 There are two different divisions in the process: one to estimate the population variance (using n - 1) and another to find the variance of the distribution of means (using n).
• 📉 T-tests are used for single samples to compare against a population mean, and also for dependent means, such as pre- and post-test scores, assuming a normal population distribution.

Q & A

What is the primary difference between the real world and the examples used in calculations regarding population characteristics?

In the real world, we often do not have information about population characteristics like the mean, variance, and standard deviation. We must estimate these values from our sample data, unlike in examples where these values are usually given.

Why is it important to make adjustments when estimating the population variance from a sample?

The sample variance is typically smaller than the population variance, which can lead to errors if not adjusted. Adjustments help to more accurately estimate the population variance and reduce the potential for error.

What is the purpose of dividing the sums of squares by 'n minus one' when calculating the estimated population variance?

'N minus one,' also known as degrees of freedom, is used to account for the fact that the sample variance is usually smaller than the population variance. It provides a more accurate estimate by considering the number of scores that can vary before the last one is determined.

How does the T distribution differ from the normal distribution, and when is it used?

The T distribution is flatter and has more scores in the extremes compared to the normal distribution. It is used when the population variance is unknown, and we need to estimate it from the sample.

What does the degrees of freedom represent in the context of T distribution?

Degrees of freedom refer to the number of scores in a sample that are free to vary before the final score is determined. It is calculated as 'n minus one,' where 'n' is the sample size.

Why is it important to choose the appropriate T distribution based on the degrees of freedom?

Each T distribution corresponds to a specific degrees of freedom, which affects the shape of the distribution. Choosing the correct one ensures accurate calculations, especially for critical values in hypothesis testing.

What role does the T distribution play when conducting a T test with a sample?

The T distribution is used as the comparison distribution when conducting a T test. It allows us to compare the sample mean to the population mean, especially when the population variance is unknown.

What is the process for calculating the standard deviation of the distribution of means in a T test?

First, estimate the population variance by dividing the sums of squares by the degrees of freedom. Then, calculate the variance of the distribution of means by dividing the estimated population variance by the full sample size. The standard deviation of the distribution of means is the square root of this variance.

What is a common application of the T test for dependent means?

A common application is in pre-post testing scenarios, where a sample is tested before and after an intervention, and the means are compared to determine if there is a significant difference.

How robust is the T test with regard to violations of the normality assumption?

The T test is very robust and can tolerate violations of the normality assumption to a significant extent. It still produces reliable results even when the population distribution is not perfectly normal.

Outlines

00:00

📚 Introduction to T-Tests and Estimating Population Variance

This paragraph introduces the concept of T-tests, which are statistical methods used when the population variance is unknown and must be estimated from a sample. It explains that in real-world scenarios, we often lack information about the population characteristics such as mean, variance, and standard deviation. Instead, we rely on sample data to make inferences about the population. The paragraph emphasizes the importance of using the sample variance to estimate the population variance, while also noting the potential for error due to the sample variance typically being smaller than the population variance. It outlines the adjustment made to the formula for estimating the population variance by dividing the sum of squares by the degrees of freedom (n-1), which accounts for the difference between the sample and population variances.

05:02

📉 Understanding T-Distributions and Conducting T-Tests

The second paragraph delves deeper into the specifics of T-distributions, which are used when estimating the population variance from a sample. It highlights that T-distributions are flatter and have more extreme scores than the normal distribution, necessitating careful selection of the appropriate T-distribution based on the degrees of freedom. The paragraph explains the process of comparing the sample mean to the population mean using the T-distribution and calculating the T-score, which involves dividing the difference between the sample mean and the hypothesized population mean by the standard deviation of the distribution of means. It also discusses the assumptions of T-tests, such as the normality of the population distribution, and the robustness of T-tests to violations of these assumptions. The paragraph concludes by reviewing the steps involved in conducting a T-test, including estimating the population variance, selecting the appropriate T-distribution, and calculating the T-score to determine whether to reject the null hypothesis.

Keywords

💡t test

A t test is a statistical hypothesis test that determines whether two sets of data are significantly different from each other. In the video, it's used to compare a sample mean to a known population mean when the population variance is unknown. The script explains that t tests are used when we estimate the population variance from our sample, which is a key step in the process.

💡population variance

Population variance is a measure of the spread of a set of data points in the entire population. The script discusses how, in real-world scenarios, we often don't have information on the population variance, so we estimate it from our sample. This estimated variance is crucial for conducting t tests.

💡sample variance

Sample variance is the variability within the data collected from a sample. The script points out that sample variance is usually smaller than the population variance, which is why adjustments are necessary when estimating the population variance from a sample.

💡degrees of freedom

Degrees of freedom in statistics refer to the number of values in the final calculation that are free to vary. In the context of the script, when calculating the sample variance, the degrees of freedom is n - 1, where n is the sample size. This concept is integral to adjusting the sample variance to estimate the population variance.

💡sum of squares (SS)

Sum of squares is the total of the squared differences between each data point and the mean. In the script, it's used as part of the formula to calculate the sample variance, which is then adjusted to estimate the population variance.

💡normal distribution

A normal distribution is a continuous probability distribution in which values are symmetrically distributed around the mean. The script mentions that the t test assumes a normal population distribution, even though we often don't know the actual distribution.

💡T distribution

The T distribution is a type of probability distribution that is similar to the normal distribution but has heavier tails, meaning it's more sensitive to outliers. The script explains that when we don't know the population variance, we use the T distribution for hypothesis testing, as it accounts for the increased uncertainty.

💡standard deviation

Standard deviation is a measure that indicates the amount of variation or dispersion in a set of values. In the video, the standard deviation of the distribution of means is calculated after estimating the population variance, which is essential for conducting t tests.

💡hypothesis testing

Hypothesis testing is a statistical method that makes a decision about a hypothesis based on data. The script describes how t tests are used for hypothesis testing, specifically to determine if there is a significant difference between the sample mean and the population mean.

💡null hypothesis

The null hypothesis is a statement of no effect or no difference that is tested with a statistical test. In the context of the script, the null hypothesis is that there is no significant difference between the sample mean and the population mean, and the t test is used to evaluate this.

💡one-tailed test

A one-tailed test is a type of hypothesis test where the alternative hypothesis specifies that the parameter is greater than or less than the value in the null hypothesis, but not both. The script mentions considering a one-tailed test with a 0.05 level of probability when determining the cutoff score on the T distribution.

Highlights

Introduction to the concept of the t-test and its importance in statistical analysis when population characteristics are unknown.

Explanation of how sample data is used to estimate population variance when actual population data is not available.

The assumption that the sample is representative of the population leads to the use of sample variance as an estimator for the population variance.

Clarification on the difference between sample variance and population variance, and the potential errors that can arise from this discrepancy.

Adjustment method for estimating population variance by using a different formula, introducing the concept of degrees of freedom.

The formula for calculating the estimated population variance, emphasizing the division by n-1 (degrees of freedom).

Understanding the concept of degrees of freedom in the context of variance calculation and its mathematical implications.

The necessity of using the t-distribution for comparison when the population variance is unknown and estimated from the sample.

Description of the t-distribution's characteristics, such as being flatter and having more extreme scores compared to the normal distribution.

Importance of selecting the appropriate t-distribution based on degrees of freedom for accurate statistical analysis.

Illustration of how the t-distribution changes with varying degrees of freedom, using a comparison with normal distribution curves.

Explanation of t-distribution tables and how they are used for determining cut-off scores in hypothesis testing.

The process of calculating the variance of the distribution of means using the estimated population variance and full sample size.

Differentiation between estimating population variance (dividing by n-1) and calculating the variance of the distribution of means (dividing by n).

The formula for computing the t-core, which is analogous to the Z-score but uses the standard deviation of the distribution of means.

Application of t-tests for single samples and dependent means, such as pre-post testing scenarios.

Assumption of the t-test regarding the normality of the population distribution and the test's robustness against violations of this assumption.

Review of the sequence and steps involved in conducting a t-test, from estimating population variance to hypothesis testing.

Transcripts

here's your lesson on a t test that will

help you with this chapter and make sure

that you are very clear on how we uh use

new information up till now whenever we

have done some calculations we've been

told what the population characteristics

are like the mean the variance and the

standard deviation in the real world we

often do not have that information so

the only information available is what

we collect in our sample if that's the

case we assume that since the sample

came from the population that it is

representative of it so then the

variance we get from our sample could be

used to estimate the variance that the

population of the population it

represents that's a great idea the

population variance is frequently

estimated from what we get in the sample

however the sample variance is usually a

little smaller than the population

variance and if there's a discrepancy

between those two things we could open

ourselves up for a lot of error like we

talked about before so we need to be

careful when we estimate the variance of

the population knowing that our sample

will probably be a little bit small so

we need to make some adjustments and

this is how we make an adjustment for

the population variance we add a

different formula into what we are used

to doing so our po our formula is going

to be the estimated variance of the

population will be the sums of the

squares divided by something a little

bit different as you recall the sums of

the squares is we take all of the scores

from our population our sample add them

up divide by the number of scores there

are and that will tell us what the mean

is and for each score we subtract the

mean from it and because those

differences those deviation scores there

will be some positives and some

negatives we want to get rid of those so

we Square each of those deviation scores

and then we add that all up when we add

that up that is what's called the sums

of the squares or SS in this formula to

account for that difference between what

the sample is and what the population is

we are going to divide by what's called

n minus one or the degrees of freedom

that's a fancy way of saying if we knew

all of the scores except for the very

last one and we knew the mean we could

mathematically figure out what the last

score is so degree of Freedom means how

many scores can vary or be different

before we can figure out what the last

score is and that is the total number

minus one we can always figure out what

the last score

is so in this case the number of scores

in the sample that are free to vary when

calculating the variance is n minus one

so we divide that sums of the squares by

the degrees of freedom and that is how

we account for the difference in the

variance

so if we've done that now we have

estimated the population variance we

needed that variance so we can also get

the standard deviation and run any

calculations like we've done before we

can now find the characteristics of the

comparison distribution or the

distribution of means as we've learned

that what we compare our sample to

depends on what we are trying to ask in

this case we need to compare to the T

distribution whenever we do not know

what the variance of the population is

and we estimate it we're going to use

the T

distribution the distribution under T is

flatter and usually has more scores in

the extremes than the normal curve so we

need to be careful that there is a

different T distribution for each

degrees of

freedom and we need to use the

appropriate one in this picture we've

usually used the normal curve up till

now T doesn't look exactly like that

unless there are extremely high number

of degrees of freedom so notice in here

we compare the normal distribution the

black line to a t distribution with

degrees of freedom of 20 so quit quick

quiz if the degrees of freedom are 20

how many people were in the

sample 21 n minus one and notice that

with 20 in the sample it looks similar

to the normal curve but take it all the

way down there's another line that shows

a t distribution with degrees of freedom

of two how many were in that

sample only three so notice how flat and

how wide the tails are with a a t

distribution with degrees of freedom of

two just like our zc scores the T

distributions have tables that have

already been calculated and since they

are flatter and there are more scores in

the extremes the cutof score of our

comparison distribution the T

distribution ution representing null has

to be accounted for so on the left hand

side you'll see the degrees of freedom

listed and each of those suggests that

well the sample size was one more of

that so in the one that's highlighted if

we look at the left hand column 15

degrees of freedom means there were 16

in the sample 15 degrees of freedom and

then at the top we look at two different

sides was this a one tail test or a

two-tail test did we have a directional

hypothesis or a n directional hypothesis

in this case we'll consider an

one-tailed or directional hypothesis and

we suggest the 0.05 level of probability

we would go to degrees of freedom of 15

one tailed at 05 probability the cut off

on our comparison distribution the T

distribution would be

1753 so that should be very similar to

what we've done before we can find the

characteristics and we can find cut offs

scores just like we've done

previously the variance of the

distribution of means is is the

estimated population variance divided by

the full sample size now this is going

to take a second to just remind

ourselves we've done this before in

chapter 5 the variance of the

distribution of means is the variance

divided by n that should sound familiar

so now we're back on track to what we've

previously done it's that we didn't know

what the sample variance was to start

with so we estimated it that was our

extra step we just added in with chapter

7 and T tests so from now on it should

look very

familiar taking the square root of that

the standard gives us the standard

deviation of the distribution of means

or the standard

error so to clarify when we are first

estimating the population variance we

divide the sums of the squares

by the degrees of freedom because we

didn't know what the variance was to

start with we had to estimate it so we

did that by dividing it by n minus one

and we estimated the population variance

later we needed to figure the variance

of the distribution of means and so we

divided our estimate by the full n there

are two different things you can slow

the video down or pause it and read the

book they are two different things it

just becomes a little tricky in there

once we have those pieces of information

then we can compute our tcore similar to

the Z the formula should look very

familiar the m is the mean of our sample

the MU is the mean of the

population and it's divided by the

standard deviation of the distribution

of means which we have now calculated

because we were able to go through the

first estimate and then figured the

variance of the distribution of

means T tests are frequently used for

single samples a sample of something and

compare it to the overall population you

may also do a t test for dependent means

this would be like a pre-post we take a

test at the beginning of the class and

get a mean we take a test at the end of

the class and get another mean and we

compare those means to see if they're

different that's a common T Test

um option that we can use the Assumption

of the T Test is that the population

distribution is normal so we often don't

know but we assume that it is normal and

the T Test is very robust we can violate

the parameters of the normal uh

population quite a bit and it still will

work very very well so let's review we

use a T Test when we don't know what the

population variance is we have to

estimate it from our sample when we

estimate the population variance from

our sample we first use the degrees of

freedom n minus one after we've done

that then we will use the comparison

distribution which is the T distribution

there is one for every degrees of

freedom there is a t table in the back

of the book that we will use depending

on our directional hypothesis and our

probability and then we will calculate

the T similarly to the other which is

our sample t-core and make the

comparison as to whether we will reject

the null

hypothesis I hope this helps you to

understand the sequence of how we do

things for a t test this is in

conjunction with your walkthrough and

your practice items