Hypothesis Testing - One Sample Proportion

Summary

In 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

00:00

📚 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.

05:02

🔍 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.

10:02

📉 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.

Keywords

💡Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population parameter based on a sample of data. It involves setting up a null hypothesis and an alternative hypothesis and then determining whether the sample data supports the alternative hypothesis over the null. In the video, hypothesis testing is used to determine if the proportion of successful students with a high school GPA is significantly different from the overall success rate.

💡Proportion

A proportion in statistics refers to the ratio of the number of successes to the total number of trials or observations. It is a way of expressing a fraction that compares part to whole. In the context of the video, the professor is testing the hypothesis about the proportion of students who are successful in their classes based on different criteria for admission.

💡Elgin Community College

Elgin Community College is the institution where Professor Dan Curler works and where the video is set. It is relevant to the video as it provides the context for the classes and the student population being discussed. The script mentions specific math classes offered at Elgin Community College and how students can get into these classes.

💡Entry-Level Math Classes

These are introductory college-level math courses that students typically take at the beginning of their academic journey. In the video, the professor lists several entry-level math classes at Elgin Community College, such as Statistics, Gen Ed Math, Math 110, and College Algebra, which are part of the educational offerings being discussed.

💡Placement Exam

A placement exam is a test used by educational institutions to determine a student's level of proficiency in a particular subject and to place them in the appropriate course. In the script, the professor discusses different ways students can get into math classes, including through SAT or CT scores, the ALEKS placement exam, or high school GPA.

💡Sample Proportion

A sample proportion is the ratio of the number of successes in a sample to the total number of observations in the sample. It is used as an estimate of the population proportion. In the video, the sample proportion of successful students with a high school GPA is calculated to be 79%, which is then compared to the overall success rate.

💡Statistical Significance

Statistical significance refers to the probability that the observed results occurred by chance. If the results are statistically significant, it means they are unlikely to have happened by random chance and can be considered reliable. The professor explains that even though the sample proportion of 79% is higher than the overall rate, it is not statistically significant due to the small sample size.

💡Z-Score

A z-score is a measure of how many standard deviations an element is from the mean. It is used in hypothesis testing to determine the probability of an event occurring by chance. In the video, the professor calculates a z-score of 1.46 to determine the likelihood of observing the sample proportion of 79% or higher.

💡P-Value

The p-value is the probability of obtaining results at least as extreme as the observed results, assuming the null hypothesis is true. A small p-value (typically ≤ 0.05) indicates strong evidence against the null hypothesis. The professor mentions a p-value of 0.072, which is used to decide whether to reject the null hypothesis.

💡Critical Value

A critical value is the value of a test statistic that determines the threshold for rejecting the null hypothesis. If the test statistic is greater than the critical value, the null hypothesis is rejected. The professor refers to the critical value method and mentions a z-value of 1.645 as the threshold for a 0.05 level of significance.

💡Statcrunch

Statcrunch is a statistical software package that can perform a variety of statistical analyses, including hypothesis testing. In the video, the professor demonstrates how to use Statcrunch to conduct a one-sample proportion test, providing a practical example of applying the concepts discussed.

💡Practical Significance

Practical significance refers to the real-world importance or meaningfulness of the results of a study, beyond just statistical significance. The professor emphasizes the importance of considering practical significance, using the example of a 6% difference in voter registration rates, which is both statistically and practically significant.

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