Stats: Hypothesis Testing (P-value Method)

Summary

This video script introduces the P Value method of hypothesis testing, contrasting it with the traditional method. It explains the fundamental concepts of hypothesis testing, including the null and alternative hypotheses, and the significance level (Alpha). The script clarifies that regardless of the method, a test statistic is calculated. The P Value method stands out by not using critical values, instead comparing the calculated P Value directly to Alpha to decide whether to reject the null hypothesis. The video uses a practical example of push pins to illustrate left, right, and two-tail tests, emphasizing that a low P Value (less than Alpha) leads to the rejection of the null hypothesis, while a high P Value (greater than Alpha) results in failing to reject it.

Takeaways

• 🔍 Hypothesis testing involves comparing two types of hypotheses: the null hypothesis (H₀) and the alternative hypothesis (H₁).
• 📉 The null hypothesis always includes an equal sign, suggesting no difference or a specific value.
• 📈 The alternative hypothesis uses a different symbol, such as 'less than', 'greater than', or 'not equal to', indicating a deviation from the null hypothesis.
• 🎯 The level of significance, denoted by Alpha (α), is a predetermined threshold used to determine the outcome of the hypothesis test.
• 📊 Hypothesis tests can be categorized into left-tail, right-tail, or two-tail tests based on the alternative hypothesis.
• 📚 The choice of test (left, right, or two-tail) depends on the wording of the claim or research question.
• 📉 The traditional method of hypothesis testing uses critical values to make a decision, whereas the P-value method does not.
• 📈 The P-value method involves calculating a test statistic and comparing the resulting P-value to the level of significance (α).
• 📝 A test statistic is calculated using specific formulas depending on the type of data and hypothesis test being conducted.
• 🔑 If the P-value is less than α, the null hypothesis is rejected, indicating support for the alternative hypothesis.
• 🔒 If the P-value is greater than α, the null hypothesis is not rejected, which means there is insufficient evidence to support the alternative hypothesis.

Q & A

What are the two types of hypotheses in hypothesis testing?

The two types of hypotheses in hypothesis testing are the null hypothesis (often denoted as H₀ or Hₙ) and the alternative hypothesis (often denoted as H₁ or Hₐ).

What does the null hypothesis typically represent in hypothesis testing?

The null hypothesis typically represents a statement of no effect or no difference, often using an equal sign (e.g., μ = 100), which is assumed to be true unless the evidence strongly suggests otherwise.

How is the alternative hypothesis represented in hypothesis testing?

The alternative hypothesis is represented using a symbol that is not an equal sign, such as less than (<), greater than (>), or not equal to (≠), indicating a deviation from the null hypothesis.

What is the significance level, or Alpha, in hypothesis testing?

The significance level, or Alpha, is the probability of rejecting the null hypothesis when it is actually true. Common values for Alpha are 0.01, 0.05, and 0.10.

What are the different types of tail tests in hypothesis testing?

The different types of tail tests are left-tail, right-tail, and two-tail tests, which depend on the directionality of the alternative hypothesis (less than, greater than, or not equal to, respectively).

What is the purpose of a test statistic in hypothesis testing?

A test statistic is a numerical value calculated from the sample data, which is used to determine the likelihood of obtaining the observed sample results under the null hypothesis.

How does the P-value method differ from the traditional method in hypothesis testing?

The P-value method does not use critical values like the traditional method. Instead, it compares the P-value, which is the probability of observing the test statistic or more extreme results, to the significance level (Alpha) to decide whether to reject the null hypothesis.

What is the P-value and how is it used in hypothesis testing?

The P-value is the probability of obtaining results at least as extreme as the observed results, assuming the null hypothesis is true. It is used to compare with the significance level (Alpha); if the P-value is less than Alpha, the null hypothesis is rejected.

What does it mean to reject the null hypothesis in hypothesis testing?

Rejecting the null hypothesis means that there is enough evidence to suggest that the alternative hypothesis is true, indicating a significant effect or difference from what was assumed in the null hypothesis.

What does it mean to fail to reject the null hypothesis in hypothesis testing?

Failing to reject the null hypothesis means that there is not enough evidence to support the alternative hypothesis, and thus the null hypothesis remains the best explanation for the observed data.

How can the concept of a 'push pin' package be used to illustrate a hypothesis test?

The 'push pin' package example illustrates a situation where the null hypothesis might be that there are 100 push pins in the package. If one suspects there are fewer (a left-tail test) or more (a right-tail test), or simply not 100 (a two-tail test), the hypothesis test would be conducted to determine if there is enough evidence to reject the null hypothesis.

Outlines

00:00

🔍 Introduction to Hypothesis Testing with P-Value Method

The video script introduces the concept of hypothesis testing, focusing on the P-Value method as opposed to the traditional method. It explains the fundamental aspects of hypothesis testing, emphasizing the two types of hypotheses: the null hypothesis (H₀), which is always stated with an equal sign, and the alternative hypothesis (H₁ or Hₐ), which uses a symbol other than an equal sign to represent different scenarios. The script also introduces the concept of the level of significance, denoted by Alpha (α), which is a critical threshold in hypothesis testing. It further explains the types of alternative hypotheses related to the direction of the test: less than for a left-tail test, greater than for a right-tail test, and not equal to for a two-tail test. The importance of understanding these concepts is highlighted through a practical example involving a package of push pins, where the number of pins is compared to the expected 100, illustrating how the choice of hypothesis affects the type of test conducted.

05:03

📊 Understanding the P-Value Method in Hypothesis Testing

This paragraph delves deeper into the P-Value method of hypothesis testing, contrasting it with the traditional method that uses critical values. The P-Value method disregards critical values and instead focuses on calculating the test statistic from given formulas, which could relate to proportions, means, or other statistical measures. The test statistic is then used to find the P-Value, which represents the area in the tail of the distribution corresponding to the test statistic. The script provides an example of a right-tail test with a test statistic of 2.61 and a P-Value of 0.0045. It explains the decision-making process in hypothesis testing: if the P-Value is lower than the predetermined level of significance (Alpha), the null hypothesis is rejected, supporting the claim of the alternative hypothesis. Conversely, if the P-Value is higher than Alpha, the null hypothesis is not rejected, indicating insufficient evidence to support the alternative claim. The script concludes by reiterating the importance of the P-Value in hypothesis testing and the implications of its comparison with Alpha for accepting or rejecting the null hypothesis.

Keywords

💡Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population parameter based on a sample. It involves formulating a null hypothesis and an alternative hypothesis and then determining if the sample data provides enough evidence to reject the null hypothesis. In the video, the concept is central to understanding the P-value method, which is a non-traditional approach to hypothesis testing.

💡Null Hypothesis (H0)

The null hypothesis is a statement of no effect or no difference that is tested using a statistical test. It is denoted by H0 or H sub zero and typically includes an equal sign to represent the status quo or the assumption of no change. In the video, the null hypothesis is the starting point for hypothesis testing, and it is always set up with an equal sign.

💡Alternative Hypothesis (H1 or Ha)

The alternative hypothesis is a statement that is contrary to the null hypothesis and represents the research hypothesis. It is denoted by H1 or Ha and uses symbols other than an equal sign to indicate a difference or effect. The video explains that the alternative hypothesis can be less than, greater than, or not equal to, which determines the type of test (one-tailed or two-tailed).

💡P-value

The P-value is the probability of obtaining results at least as extreme as the observed results, assuming that the null hypothesis is true. It is used in hypothesis testing to determine the strength of evidence against the null hypothesis. The video focuses on the P-value method, where the P-value is compared to a predetermined level of significance (Alpha) to make a decision.

💡Level of Significance (Alpha)

The level of significance, denoted by Alpha, is the threshold used to determine whether to reject the null hypothesis. Common levels of significance are 0.01, 0.05, and 0.10. In the video, Alpha is the criterion against which the P-value is compared to decide whether there is enough evidence to reject the null hypothesis.

💡Test Statistic

A test statistic is a value calculated from sample data that is used to make a decision in hypothesis testing. It quantifies the degree to which the sample data contradict the null hypothesis. The video mentions that both traditional and P-value methods require the calculation of a test statistic, which can vary depending on the type of data and hypothesis test being conducted.

💡Critical Value

A critical value is a threshold value used in traditional hypothesis testing to determine the rejection region. If the test statistic falls into this region, the null hypothesis is rejected. The video contrasts the use of critical values in traditional methods with the P-value method, which does not use critical values.

💡One-Tailed Test

A one-tailed test is a type of hypothesis test where the alternative hypothesis specifies a direction of the effect or difference. It has only one tail in the distribution, either to the left or right, indicating the direction of the expected effect. The video uses the example of a left-tail test, where the alternative hypothesis is less than the null hypothesis.

💡Two-Tailed Test

A two-tailed test is a hypothesis test where the alternative hypothesis does not specify a direction of the effect or difference. It has two tails in the distribution, one to the left and one to the right, indicating that the effect could be in either direction. The video explains that a two-tail test occurs when the alternative hypothesis is not equal to the null hypothesis.

💡Rejection Region

The rejection region is the area under the probability distribution where the null hypothesis would be rejected. It is determined by the critical values in traditional hypothesis testing. The video does not explicitly mention the rejection region, but it is implied when discussing the comparison of the test statistic to the critical values.

💡Type I and Type II Errors

Type I error occurs when the null hypothesis is true but is rejected, while Type II error occurs when the null hypothesis is false but is not rejected. The video does not explicitly mention these terms, but understanding them is crucial for interpreting the results of hypothesis tests. The level of significance (Alpha) controls the probability of a Type I error.

Highlights

Introduction to the P Value method for hypothesis testing, an alternative to the traditional method.

Explanation of the two types of hypotheses: null hypothesis (H0) and alternative hypothesis (H1 or Ha).

The null hypothesis always uses an equal sign, while the alternative hypothesis uses a different symbol.

Introduction of Alpha, the level of significance in hypothesis testing.

Three popular levels of significance: 0.01, 0.05, and 1%.

Different types of alternative hypotheses: less than, greater than, or not equal to.

Tail tests: left tail, right tail, and two-tail tests based on the alternative hypothesis.

Practical example using a package of push pins to illustrate left and right tail tests.

The importance of the wording of the alternative hypothesis in determining the type of test.

Comparison between the traditional method and the P Value method for hypothesis testing.

The P Value method does not use critical values, unlike the traditional method.

Process of finding the test statistic for both traditional and P Value methods.

Explanation of how to calculate and interpret the P Value in hypothesis testing.

The P Value is the area under the curve that is more extreme than the test statistic.

Comparing the P Value to the level of significance (Alpha) to make a decision.

If the P Value is lower than Alpha, the null hypothesis is rejected.

If the P Value is higher than Alpha, the null hypothesis is not rejected (fail to reject).

The implications of rejecting or failing to reject the null hypothesis in terms of supporting the alternative hypothesis.

Clarification of the terminology used in hypothesis testing: reject, fail to reject, and support the claim.

Transcripts

all right in this video I want to show

you the basics of hypothesis testing for

the non-traditional method or the P

Value method to be more specific all

right the P Value method I have shown in

a separate video um the basics of

hypothesis testing using the traditional

method so there are two different kinds

um actually probably more than two but

I'm only showing you two in my videos so

real quickly here the basics of

hypothesis testing no matter what method

you're using traditional or P value is

that there there are going to be two

types of hypotheses right there's the

null hypothesis and there's the

alternative

hypothesis let's see almost every book

I've used uses H subz or H KN as the

null hypothesis and no matter what uh

parameter you're talking about whether

it's a PO uh whether it's a proportion

or a mean or a standard deviation the

null hypoth hthis always uses an equal

sign right it always has an equal sign

in it the alternative hypothesis maybe

your book uses H sub one or H sub a

whatever the case may be always use as a

symbol that is not an equal sign right

so in this case it's a less than symbol

or a greater than symbol or a not equal

to symbol but no matter what type of

hypothesis testing it is you will always

be given some level of significance here

okay so whether it's 01 or 05 or P

something like that now that level of

significance is called Alpha right it's

called Alpha which is the Greek uh

lowercase letter A here that's the first

letter of the Greek alphabet so so and

we're not restricted to 015 or 10% or

something like that it could be any

other type of level of significance but

these are the three more popular

ones okay the other thing I want to tell

you about and keep your eye on these

three alternative hypotheses here right

so less than greater than or not equal

to is depending on what kind of a test

we're talking about you might have a

left tail test a right tail test or a

two-tail test and that all depends on

the alternative hypothesis all right so

I'm using H sub one as my alternative

hypothesis notation here but if the

alternative hypothesis is less than we

have a left tail test if it's greater

than we have a right tail test or if

it's not equal to we have a two-tail

test we have two tail test so that is

also important and significant here and

what do I mean by that I didn't show

this in my previous video but for

example here let's say I've got let's

see I got a little package here of push

pins can you guys see that I think you

can see that in the video here now all

right and it says that there should be

100 push pins in here now if I think I

got jipped right and they're probably

I'm thinking they're putting less than

100 push pins in there then less than

would be a left tail test all right or

maybe I'm thinking there being generous

as I buy these push pins and they're

actually putting more than 100 in well

more than would be a right tail test

greater than okay where if I State the

claim right all of these are claims here

if I State the claim as um I don't think

they're putting 100 in there right

notice I'm not specifying when I say I

don't think they're putting 100 in there

I believe there aren't 100 in there I'm

not specifying whether it's less than

100 or greater than 100 so in that case

it would be a two-tail test so depending

on the wording that you see um it could

be one of those types of alternative

hypotheses or claims okay so what I

showed again in a previous video you

could look it up is the traditional

method and what I'm going to show you in

this video is the P Value method right

but notice I wrote on this little piece

of paper here that both methods doesn't

matter which method you're using both

method methods require you to find a

test statistic so both meth methods use

a test statistic whether your test

statistic is is uh is with proportions

that would be P hat minus p over the

square < TK of uh Little P * Q Over N or

maybe you're you're dealing with with uh

sample means and population means and so

your test statistic would be with this

formula here so you probably are

familiar with those formulas okay but

regardless of the method both of them

use test statistics okay so what the P

value does

separately from the traditional method

is this all right the traditional method

uses something called critical

values right I'll just put this on here

real quick the traditional method so

traditional method uses something called

critical values well guess what and I'm

going to cross that out the P Value

method does not use that at all all

right P Value method says look I don't

really care about the critical value

whatsoever so that's kind of a nice

little thing here so let's say that for

example we were dealing with a right

tail test right so I'll draw a little

right tail here right so let's say for

example we were dealing with the right

tail test that's our um H sub one is a

greater than then what the P value says

is this all right what the P value says

is this what we're going to do is we're

going to find using uh the formulas that

I just showed you a second ago we're

going to

find put here find we're going to find

our test

statistic all right we're going to find

our test statistic using that formula

you just saw in the previous page and

then we're going to put that in I'm

going to write this as uh how about TS

all right test for statistic we're going

to we're going to put that in here and

let's this is a right tail

test um let's say our test statistic

here I don't know I'm just kind of

making something up but let's say our

right our test statistic came out to be

something like uh how about

-

2.61 for example all right sorry

positive this is on the right hand side

so let's say our test statistic came out

to

2.61 okay so for example here 2.61 now

do you see that um this area the area

sitting over here to the far right of

this is

0.0045 okay you can look that up this

area is

0.0045 right it's

0045 and you can look that up on a table

if you want to or you can use um Excel

to figure that that area out but here's

here's the here's the deal that area is

what's called our P value all right so

this area is the P value now all right

so the area sitting over here to the

right in this case of a test statistic

is our P value okay so that area in this

case

0045 is our P value that we are going

going to compare right we're going to

compare the P

value to our Alpha to our level of

significance and that's how P value uh

method or the testing works so in our

case I have a P value

0.0045 and let's say that we had an

alpha let's say that we were looking for

a level of significance of

0.001 for example I don't know we'll

pick on 0.01 all right so our Alpha our

P value is sitting right here our Alpha

is sitting right here of

0.01 and as you compare these two do you

notice that this one here our P value is

less than

01 all right and there's a phrase that

goes along with this if the P value is

low right low lower than my Alpha my

level of significance then the null must

go so we're going to take the null

hypothesis and we're going to reject it

right we're going to reject it what if

though let's just say for instance what

if our P value was greater than our

Alpha right I know this is not the case

in this particular case but let's say we

had a P value of

0.0

28 n or something like that all right

and in this particular case look our

Alpha I mean our P value rather would be

bigger than our Alpha of 0.01 so if this

is the case if your if your P value is

high right is larger than your Alpha

then the null must fly so we're going to

keep all right the null

hypothesis so hypothesis testing is all

about the null we either reject it or we

in my case I put keep here but

technically ter the term is we fail to

reject okay so this is a reject here if

the uh P value is low and if the P value

is high then the null must fly what

we're really doing is we are really

failing to reject that's what's going on

here when I say the words keep okay

we're failing to reject so if the null

is if the P value is low the null must

go if the P value is high the null flies

and what that really means is if we

reject the null hypothesis what we're in

essence saying is we

support the claim of the alternative

hypothesis all right if we keep the null

hypothesis what we're in essence saying

is this may be true therefore we don't

have enough evidence to support our

claim right so the wording of this is

pretty tricky you've got to get your

mind around that as well but this is how

P value testing works