How to calculate One Tail and Two Tail Tests For Hypothesis Testing.

Summary

This tutorial explains one-tailed and two-tailed hypothesis tests, using a normal bell curve to illustrate the distribution of possible events. The population mean is denoted by the Greek letter 'mu', and the sample mean can vary within a certain range. A one-tailed test with a 95% confidence level implies a 5% chance (Alpha) of the sample mean falling in the rejection region. In contrast, a two-tailed test divides Alpha by two, placing 2.5% in each tail, ensuring a 95% chance of the sample mean falling within the acceptable range. Critical values for a 95% two-tailed test are ±1.96, while for a one-tailed test, they are ±1.645. The video promises more in-depth statistical insights.

Takeaways

• 📊 The tutorial discusses one-tailed and two-tailed hypothesis tests, which are statistical methods to make inferences about a population.
• 🔵 A normal bell curve is used to represent all possible events, with the population mean (μ) at the center.
• 🌿 The sample mean (x̄) can vary and is represented within a green area on the bell curve, indicating its possible positions.
• 🎯 In a one-tailed test with a 95% confidence level, there's a 5% chance (Alpha) that the sample mean will fall in the red, or rejection, region.
• 🚫 The red area signifies the rejection region where the null hypothesis would be rejected if the sample mean falls within it.
• 🔄 For a one-tailed test, the red area can be to the right or left of the mean, depending on the direction of the hypothesis.
• 🔁 In a two-tailed test at a 95% confidence level, the 5% Alpha is split into 2.5% in each tail, providing a balanced test for deviations in either direction.
• 🔢 The critical values for a two-tailed test at 95% confidence are ±1.96 z-scores, indicating the boundaries of the rejection regions.
• 🔄 Switching back to a one-tailed test, the critical values adjust to ±1.645 z-scores, reflecting a higher probability of the sample mean being extreme in one direction.
• 📚 The tutorial serves as an introductory lesson on hypothesis testing, with more advanced statistical concepts to be explored in subsequent lessons.

Q & A

What does the bell curve represent in the context of the tutorial?

The bell curve represents 100% of all possible events, with the population mean at the center.

What is the Greek letter used to denote the population mean?

The Greek letter used to denote the population mean is 'mu', which looks like a 'u' but is pronounced as 'me'.

What is the notation used for the sample mean in the tutorial?

The notation used for the sample mean in the tutorial is 'x-bar' (x̄).

What is the probability range for the sample mean in a one-tailed test with a 95% confidence level?

In a one-tailed test with a 95% confidence level, there is a 5% chance the sample mean will show up in the red area, which is also known as Alpha.

What is Alpha in the context of hypothesis testing?

Alpha is the probability of the sample mean showing up in the red area, which is also known as the rejection region. It can also be referred to as the P-value.

How is the red area distributed in a two-tailed test at a 95% confidence level?

In a two-tailed test at a 95% confidence level, the red area is split equally, with 2.5% on each tail (left and right).

What is the significance of the 2.5% on each tail in a two-tailed test?

The 2.5% on each tail represents the probability of the sample mean falling in the rejection region for a two-tailed test at a 95% confidence level.

What are the critical values for a two-tailed test at a 95% confidence level?

The critical values for a two-tailed test at a 95% confidence level are z-scores of -1.96 on the left and +1.96 on the right.

How do the critical values change for a one-tailed test?

For a one-tailed test, the critical value on the right is a positive 1.645, and on the left, it would be a negative 1.645.

What is the purpose of discussing one-tailed and two-tailed tests in the tutorial?

The purpose is to introduce and explain the concepts of one-tailed and two-tailed hypothesis tests, including their setup, probabilities, and critical values.

What does the term 'rejection region' refer to in hypothesis testing?

The 'rejection region' refers to the area of the distribution where the sample mean would lead to the rejection of the null hypothesis.

Outlines

00:00

📊 Introduction to One-Tail and Two-Tail Tests

This paragraph introduces the concepts of one-tail and two-tail tests in statistical analysis. The speaker begins by explaining the setup using a normal bell curve, which represents 100% of all possible events. The population mean, denoted by the Greek letter mu (µ), is situated at the center of the curve. The sample mean, denoted by x̄, can vary and is represented within a green area on the curve. The paragraph discusses how the sample mean can fall above or below the population mean, and the probability associated with its occurrence. The speaker then explains the one-tail test with a 95% confidence level, where there is a 5% chance (represented by the red area) that the sample mean will fall outside the green area, known as the rejection region or alpha. The concept of the P-value is also introduced. The paragraph concludes with a transition to a two-tail test, where the red area is split equally on both sides of the curve, representing a 2.5% chance in each tail. The critical values for a 95% confidence level are also mentioned, with -1.96 and 1.96 for a two-tail test, and -1.645 and 1.645 for a one-tail test.

Keywords

💡Normal Bell Curve

The normal bell curve, also known as the Gaussian distribution, is a fundamental concept in statistics representing the distribution of a variable that is normally distributed within a population. In the script, the bell curve symbolizes the entire range of possible events, with the peak at the center representing the population mean (μ). It is used to visualize the likelihood of different sample means occurring in relation to the population mean.

💡Population Mean (μ)

The population mean (μ) is the average value of a population's data set. It is a theoretical value that represents the central tendency of all possible measurements. In the script, the population mean is depicted as the center of the normal bell curve, indicating that it is the expected value around which the data is symmetrically distributed.

💡Sample Mean (x̄)

The sample mean (x̄) is the average of a subset of data drawn from a larger population. It is denoted by the script as a value that can fall anywhere within the green area of the bell curve, signifying that it is an estimate of the population mean. The sample mean is crucial for statistical inference, as it is used to make inferences about the population from which the sample was drawn.

💡Probability

Probability in the context of the script refers to the likelihood of a sample mean occurring within a certain range of the population mean. The tutorial discusses setting up a test with a 95% probability that the sample mean will fall within the green area of the bell curve, which implies a 5% chance (alpha) that it will fall in the red area, indicating a significant deviation.

💡Alpha (α)

Alpha (α) is the probability of rejecting the null hypothesis when it is actually true, also known as the significance level. In the script, alpha is represented by the red area of the bell curve, which signifies the rejection region. A 5% alpha level is mentioned, indicating a 5% risk of incorrectly rejecting the null hypothesis.

💡P-value

The P-value is the probability of observing data as extreme or more extreme than what was actually observed, assuming the null hypothesis is true. It is mentioned in the script as another term for alpha and is used to decide whether to reject the null hypothesis. A small P-value suggests strong evidence against the null hypothesis.

💡Rejection Region

The rejection region is the area of the distribution where the sample mean would fall if the null hypothesis is false, leading to its rejection. In the script, the red areas to the far left and right of the bell curve are the rejection regions for a two-tailed test, while the red area on one side represents the rejection region for a one-tailed test.

💡One-tail Test

A one-tail test, also known as a one-sided test, is a type of hypothesis test where the rejection region is entirely on one side of the distribution. The script explains setting up a one-tail test with a 95% confidence level, where there's a 5% chance the sample mean will fall in the red area, suggesting an extreme outcome in one direction.

💡Two-tail Test

A two-tail test is a hypothesis test where the rejection region is split between both tails of the distribution. The script describes dividing the alpha (5%) by two, placing 2.5% in each tail, which means there's a 2.5% chance the sample mean will fall in either the lower or upper tail, indicating an extreme outcome in either direction.

💡Confidence Level

The confidence level is the percentage of times a sample mean would fall within a certain range of the population mean if an infinite number of samples were taken. In the script, a 95% confidence level is used, which means there's a 95% chance that the true population mean lies within the range defined by the sample mean and the critical values.

💡Critical Values

Critical values are the points on the distribution that define the boundary of the rejection region. In the script, critical values for a two-tailed test at a 95% confidence level are given as ±1.96, while for a one-tailed test, they are ±1.645. These values help determine whether the observed sample mean is significantly different from the hypothesized population mean.

Highlights

Introduction to one-tailed and two-tailed tests

Explanation of the normal bell curve representing 100% of all possible events

Definition of population mean (mu) and its representation on the bell curve

Description of the sample mean (x̄) and its potential positions relative to the population mean

Probability of the sample mean falling within the green area in a one-tailed test

Use of a 95% probability for the sample mean to fall within the green area

Explanation of the red area as the rejection region (Alpha) with a 5% chance

The concept of Alpha as a probability and its role in hypothesis testing

Setting up a one-tailed test with the red area to the right of the population mean

Probability of the sample mean appearing in the red area in a one-tailed test

Introduction to the two-tailed test and its 95% confidence level

Division of Alpha into two equal parts for the two-tailed test

Coverage of all observations by the green and red areas in a two-tailed test

Critical values in a two-tailed test at the 95% confidence level

Identification of critical z-scores for the two-tailed test

Return to the one-tailed test and its critical values

Conclusion and anticipation of further statistical learning

Transcripts

in this tutorial I'm going to discuss

one tail and twail

tests we set this up by drawing a normal

bell

curve and the bell curve represents 100%

of all possible

events right in the middle is the

population mean and we use the Greek

letter

mu it looks kind of like a u but we call

it

me a sample mean can be anywhere within

this green area

for sample mean I use the notation

xar the value of the sample mean can be

above the population mean it can also be

below the population

mean and there's a probability that it

will show up somewhere the sample mean

will be somewhere in this green

area with a Oneil test there's some

probability that the sample mean will be

in the green

area and I'm going to use a 95%

probability and that means there's a 5%

chance the sample mean will show up in

the red

area this red area is often known as

Alpha the Greek letter

Alpha also it can be used as the P

value it's typically called the

rejection

region the sample mean has to show up in

the red area or in the green area

because these two add up to

100% in this example I'm using a 95%

confidence and you could actually use

any level you wanted

to I can also have a one tail test with

the red area to the right of the

population

mean there's a 95% chance a randomly

selected mean will be in the green area

sample

mean there is a 5% chance it will appear

in the red area a randomly selected

sample

mean but there is a 95% chance it will

be in the green

area again the red area is often called

Alpha refer to as alpha or the P value

and we refer to it also as the rejection

region now when I create a two-tail test

at a 95% level of

confidence I will take that red

area and I'll put half of it to the

right and half of it to the

left in other words I take Alpha /

two so I take the

5% divided by two and I put

2.5% on the left side and 2.5% on the

right

side you'll also see this Alpha / two in

other statistics formulas and this is

what it's referring to

there is 2.5% on the left and

2.5% on the

right 2.5% plus

95% plus

2.5% equals 100% this means all

observations and all events are

covered there is a 95% chance the sample

mean will be in the green area a

randomly selected sample

mean there is a 2.5% chance the sample

mean will be in the bottom

tail and there's a 2.5% chance the

sample mean will be in the upper tail

right

there now at a 95% level there's

critical values and these are zc

scores negative 1.96 on the left and

positive 1.96 on the right

when I return to a one tail test the

critical value on the right is a

positive

1.645 and the critical value to the left

would be a negative

1.645 and this has just been an

introduction to one and twoa

tests and there's more to come more to

learn about stats