Hypothesis Testing - One Sample Proportion
Summary
TLDRIn this educational video, Professor Dan Curler from Elgin Community College introduces hypothesis testing for proportions. He discusses entry-level math classes and explores whether high school success is predictive of college performance. Using a sample of 63 students, he demonstrates the hypothesis testing process, including setting up null and alternative hypotheses, calculating the test statistic, and interpreting the p-value and z-score. The video also covers practical significance versus statistical significance, emphasizing the importance of context in interpreting results.
Takeaways
- 🎓 Professor Dan Curler introduces a video on hypothesis testing about a proportion in a statistics series.
- 📚 Elgin Community College offers various entry-level math classes including statistics, gen ed math, math for elementary educators, and college algebra.
- 📈 The college has different pathways to enter math classes, such as Math 095, Math 98, Math 99, SAT or ACT scores, and recently, high school GPA.
- 🤔 The video poses a question about whether students with high school success are more likely to succeed in college, focusing on a specific group admitted based on high school GPA.
- 📊 A sample proportion of 79% success rate is compared to an overall college success rate of 71%, prompting a hypothesis test.
- 📘 The criteria for performing a hypothesis test are explained, including the requirement that n*p*(1-p) should be at least 10 and the sample should be less than 5% of the population.
- 📉 The mean and standard deviation of the sample proportion are calculated, setting the stage for hypothesis testing.
- 🔢 A z-score of 1.46 is computed, indicating how many standard deviations the sample proportion is from the population proportion.
- 📝 The six steps of hypothesis testing are outlined, including defining hypotheses, determining alpha, computing the test statistic, finding the p-value and critical value, making a decision, and drawing a conclusion.
- 🚫 The conclusion for the first example is that there is not enough evidence at the 0.05 level to support the claim that the proportion is more than 0.71, due to a small sample size.
- 📊 StatCrunch software is demonstrated for conducting hypothesis tests, including how to input data and interpret results.
- 🗳️ A second example involving voter registration rates among children of immigrants is presented, showing a statistically significant difference compared to the general population.
- 🤔 The importance of distinguishing between statistical significance and practical significance is highlighted, emphasizing the need to consider the meaningfulness of results.
Q & A
What is the main topic of Professor Dan Curler's video?
-The main topic of the video is hypothesis testing about a proportion in statistics.
What are the four entry-level college math classes mentioned?
-The four entry-level college math classes mentioned are statistics, general education math, math for elementary educators (Math 110), and college algebra.
What are the different ways students can get into these math classes?
-Students can get into these math classes through Math 095 (preparation for gen ed math), Math 98 (intermediate algebra), Math 99 (combined beginning and intermediate algebra), SAT/ACT scores, ALEKS placement exam, or using their high school GPA.
What was the success rate for students who entered using only their high school GPA and a fourth year of math?
-The success rate for students who entered using only their high school GPA and a fourth year of math was 50 out of 63 students, or 79%.
What is the population proportion used for comparison in the hypothesis test?
-The population proportion used for comparison in the hypothesis test is 71%.
What is the z-score and p-value obtained in the hypothesis test?
-The z-score obtained in the hypothesis test is 1.46, and the p-value is 0.072.
What is the critical value for the test?
-The critical value for the test, with an alpha level of 0.05, is 1.645.
What conclusion did Professor Curler reach regarding the hypothesis test?
-Professor Curler concluded that there was not enough evidence at the 0.05 significance level to support the claim that the proportion is more than 71%, meaning the sample proportion of 79% was not statistically significant.
What is the distinction between statistical significance and practical significance?
-Statistical significance means that the result is unlikely to have occurred by chance, while practical significance refers to whether the result has meaningful real-world implications. For example, a small difference in proportions may be statistically significant but not practically meaningful.
What tool does Professor Curler use to perform the hypothesis testing?
-Professor Curler uses StatCrunch to perform the hypothesis testing about proportions.
Outlines
📚 Introduction to Hypothesis Testing for Proportions
Professor Dan Curler from Elgin Community College introduces a video on hypothesis testing for proportions. He begins with an overview of the math classes available at the college and the various pathways to enroll in them, including the use of high school GPA as a placement criterion. The professor then presents a scenario where he investigates whether students with high school success are more likely to succeed in college. Using a sample of 63 students with a success rate of 79%, he compares it to the overall college success rate of 71%. The video explains the criteria for conducting a hypothesis test, which includes ensuring the sample size is large enough and represents less than 5% of the population. The professor demonstrates how to calculate the mean and standard deviation for the sample proportion and how to determine the p-value and z-score for the test.
🔍 Hypothesis Testing Steps and StatCrunch Application
Continuing the discussion on hypothesis testing, Professor Curler outlines the six steps involved in the process, emphasizing the importance of defining null and alternative hypotheses, selecting an alpha level, computing the test statistic, and determining the p-value or critical value. He uses the example of the sample proportion of 79% not being statistically significant compared to the 71% college-wide rate due to the small sample size. The professor then demonstrates how to perform a hypothesis test using StatCrunch software, showing how to input data, set up the test, and interpret the results, including the calculation of the z-test statistic and p-value. He also discusses another example involving voter registration rates among children of immigrants compared to the general population, highlighting the difference between statistical significance and practical significance.
📉 Understanding Statistical vs. Practical Significance
In the concluding part of the video, Professor Curler emphasizes the distinction between statistical significance and practical significance. He illustrates this with the voter registration example, where a 6% difference between the sample statistic and the comparison rate is both statistically and practically significant. The professor advises viewers to consider the meaningfulness of their results, questioning whether a statistically significant result has practical implications. He wraps up the video by inviting viewers to subscribe for more content on hypothesis testing and thanks the Elgin Community College Board of Trustees for supporting his sabbatical, which enabled him to create the video series.
Mindmap
Keywords
💡Hypothesis Testing
💡Proportion
💡Elgin Community College
💡Entry-Level Math Classes
💡Placement Exam
💡Sample Proportion
💡Statistical Significance
💡Z-Score
💡P-Value
💡Critical Value
💡Statcrunch
💡Practical Significance
Highlights
Professor Dan Curler introduces a video on hypothesis testing about a proportion.
Discusses math classes at Elgin Community College and how to enroll in them.
Explains the prerequisites for hypothesis testing using the normal distribution.
Calculates the mean and standard deviation for the sample proportion.
Determine the p-value and z-score for the sample proportion of 79%.
Outlines the six steps of hypothesis testing, including defining hypotheses and determining alpha.
Uses a real-world example involving high school GPA and college success rates.
Analyzes the sample size and its statistical significance in hypothesis testing.
Demonstrates how to perform hypothesis testing using StatCrunch software.
Presents an example from the US Census on voter registration rates.
Explains the difference between statistical significance and practical significance.
Concludes that there is not enough evidence to support the claim that the proportion is more than 0.71 based on the sample.
Shows how to use StatCrunch for one-sample proportion tests with summary data.
Discusses the importance of considering the practical implications of statistical results.
Provides a method to perform hypothesis testing for proportions in StatCrunch with data exclusions.
Concludes that there is enough evidence to support the claim that the proportion of registered voters is different for children of immigrants.
Emphasizes the importance of understanding the practical significance of statistical results beyond just statistical significance.
Transcripts
hello this is professor dan curler of
elgin community college back with
another video in my statistics series in
this one we're going to dive into the
specifics of hypothesis testing about a
proportion okay let's get to it
[Music]
we're going to start off today actually
talking about elgin community college
and some of the math classes we have and
how you can get into those classes and
then we'll do a little hypothesis
testing so we have four kind of
entry-level college-level math classes
we have statistics
gen ed math we have math 110 which which
is math for elementary educators and
then we have college algebra there's a
variety of ways you can get into these
math 095 is basically preparation for
gen ed math so that just gets you into
102 and 104. math 98 is intermediate
algebra that can get you into all of
these there's also math 99 which is a
combined beginning and intermediate
then there's you can get an sat or a ct
score that's high enough same thing on
the aleks placement exam and then
relatively recently we added this high
school gpa which was a statewide
initiative there was legislation that
was passed we had been investigating
this as well we know students that do
well in high school are likely to do
well in college as well so what if i
wonder if that last group who did well
in high school actually is more likely
to succeed in college than the other
groups that were in there
well what we could do is we could look
at the overall success rate in these
classes it's about 71 percent and then
we have a sample of students who've come
into ecc and they got into those classes
without any placement exam without any
set or act they just had their high
school gpa score plus you had to have um
a fourth year math class as well and of
those 50 out of 63 were successful in
one of those four classes that they took
this was in a single semester i believe
it can't remember exactly when this was
implemented might have been fall 2020.
so we have a sample proportion then is
79 percent clearly higher but only 63
maybe it's not statistically higher
so let's test it and we'll go through
this hypothesis testing process remember
this only works we can only do this
hypothesis test
if we have this criteria met that n
times p
times 1 minus p is at least 10 and we
have less than 5 percent of the
population if those conditions are met
then we'll fit this normal distribution
so we have 63 here the p proportion will
be the 0.71 plug those in yes that is at
least 10 and we do have less than 5
percent of all possible students taking
these classes
okay so if those conditions are met then
the mean of the sample proportions will
be the same as the population proportion
and the standard deviation will be
square root of p times 1 minus p all
over n fill in those values we get 0.71
is our proportion and we get a standard
deviation about 0.0572
and now that we have the mean 0.71 and
the standard deviation we can look at
our value that we have here we have 71
percent that we're comparing with and
then 50 or 79 percent is our sample we
can figure out where those go on the
curve so 50 over 63 is over here the
p-value be the probability of getting
that value or more extreme in this case
that's about 0.072
we could also compute a z-score here 50
over 63 minus the p over the standard
deviation we get about 1.46
that's a z-score we can treat that as a
z and do the same thing this would be if
we wanted to do the critical value
method to do the critical value method
we need the value that has .05 to the
right or whatever our alpha is in this
case z point zero five is one point six
four five
so now let's go through those six steps
we have to first define the null and
alternative hypotheses our null
hypothesis is that the proportion is the
same as it was for the other groups for
the alternative remember there are three
possibilities here so here's the
distribution under the null hypothesis
now we could suspect hey do we think
it's greater than that
or do we think it's less than that are
we not sure could it be less than or
greater than so we put it not equal to
so there's three possibilities for the
alternative
in our case we were wondering i was
wondering if these students did better
so i was wondering if the success rate
was higher so greater than for this
particular example
now we have the alpha determine alpha
the level of significance a good default
choice here is 0.05 you don't have to
choose that uh determine the compute the
test statistic so that's this z stats
test statistic in this case you take the
sample minus its mean which would be the
population proportion and divide by the
standard deviation and that's the 1.46
for the p-value you're just going to
find the probability of being to the
right of that 50 over 63 and that's
0.072
for the critical value we need to
convert these to z's so we have our 1.46
we need to find the z with 0.05 to the
right
and that was
1.645
all right we have those two in there now
now we need to make our decision do we
reject the null hypothesis or not
in both cases we look at the p value
it's not above our not below our
threshold and the 1.46 is not in the
critical region it's not above 1.645
so that would mean we do not reject the
p equals 0.71 now this is really
important
i'm using the language we're not going
to reject the null hypothesis
the null hypothesis that the proportion
for this group is also 0.71 might not be
correct it might be 0.72 0.74 it might
even be 0.84
we don't know
all we can say is that we're not going
to reject that it's 0.71
okay that's all we can say and when we
have our conclusion we say okay there's
not enough evidence at the 0.05 level to
support the claim that the proportion is
more than 0.71
so we had 79 was our sample proportion
but because we only had a small sample
size of 63 that wasn't statistically
significant we do not have enough
evidence to say that it's higher than 71
percent
all right let's talk about how to do
this in statcrunch this is going to be
stat proportion stats one sample and
here we're going to do with summary it
feels like we have data but we actually
just have the summary we have that 50
out of 63. so now we go and we enter in
our counts 50 out of 63 for the
proportion we're going to do we're going
to compare to 71.71
and then we'll hit compute and we'll see
our p value in fact we'll see our z test
statistic there as well here's another
example i found this information from
the us census that of those who are
eligible to vote 71 percent were
registered to vote for the nas the last
presidential election
so we might wonder we have our children
of immigrants database and i wonder if
the proportion of those who are eligible
to vote
who registered is different now it's
important to note we have their
citizenship status they're not all
eligible to vote we have 84 percent of
them are eligible to vote so the
question we're going to ask we're going
to ask this before looking at the data
is is the proportion of eligible voters
who are registered different for
children of immigrants than for the
general population so pay attention to
that phrasing if we look at actually i
made a graph here for the proportion of
those who are eligible who are
registered and it's actually 77 percent
now keep in mind we've talked about this
before we should read the details of
this database and how these data were
collected it's possible that this sample
doesn't represent all children of
immigrants we had a much higher
education rate but it it we the only
thing we can do is take it at face value
so we have a 77 voter registration
rate for those who are eligible to vote
in this children of immigrants database
let's do this in statcrunch this is a
little tricky so we're going to go to
stat proportion stats one sample with
data
the variable we want to look at here is
registered so we'll scroll all the way
down but this is a little tricky we want
to just pick those who are eligible so
we have to exclude those who are not
citizens and we can do that where
there's this where box so we'll go in
we're going to build a formula
and we're going to build a formula where
if you scroll down that's their current
citizenship so we want citizen now
is not equal to so it's a little where
it's an exclamation point and equal to
is not equal to
and then not a citizen
okay so a little tricky there but then
we can do our null hypothesis uh p
equals 0.7 alternative will be not equal
to and then we'll go down and hit
compute
and we have our results so we have our
test statistic pretty high here
7.51 for the p value we'll go take a
look at the statcrunch output again
and we can see with a z of 7.51
you're going to that's basically off the
scale remember three standard deviations
each way is 99.7 percent so we're just
going to have p value less than .001
so here we did have an extreme
observation
so we would reject the null hypothesis
so our conclusion then
is in this case there is enough evidence
at the 0.05 level of significance to
support the claim that the proportion of
eligible voters who are registered is
different for children of immigrants so
in this case there is enough evidence to
support our alternative claim one last
little note i want to make here it's
important to understand the difference
between statistical significance and
practical significance you have to look
at your sample statistic and the one
you're comparing with and just because
they're statistically significant
doesn't mean that it has any practical
meaning in this case we had a sample
proportion of eligible voters who were
registered with 77 percent and the
comparison was 71 percent in my opinion
that difference of six percent is pretty
significant that has some meaning it was
statistically significant but it seems
to be practical whereas if you had a
difference between 71 percent and 72
percent yeah it might be statistically
significant but does it actually have
any meaning so be sure you're kind of
thinking deeply about your results and
just because you get a statistically
significant result does it actually mean
anything is there actually
is a is there actually a meaningful
difference between those all right that
is it for this video on hypothesis
testing about proportions i hope this
was helpful if you're interested in
seeing more of these you can subscribe
hit the bell to get notified we've got a
whole series of these coming out a bunch
more about different hypothesis tests as
always thank you to the elgin community
college board of trustees who approved
my sabbatical for the spring 2021
semester and that's how i was able to
record all these videos for you and
thank you so much for watching i will
see you in the next one
you
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