t-tests mini lecture
Summary
TLDRThis 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
📚 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.
📉 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.
Mindmap
Keywords
💡t test
💡population variance
💡sample variance
💡degrees of freedom
💡sum of squares (SS)
💡normal distribution
💡T distribution
💡standard deviation
💡hypothesis testing
💡null hypothesis
💡one-tailed test
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
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