What is p-value? How we decide on our confidence level.
TLDRThe video script presents a conversational exploration of the concept of p-values and hypothesis testing through the example of a coin toss experiment. The discussion begins with a basic explanation of the null hypothesis, which assumes the fairness of a coin. As the experiment progresses, the probability of getting consecutive heads decreases, leading to the calculation of p-values and the introduction of the significance level (alpha). The script illustrates that a p-value is the probability of observing an outcome as extreme as, or more extreme than, the one observed if the null hypothesis were true. The significance level is set based on the point at which one would be comfortable rejecting the null hypothesis. The video concludes with the understanding that if the p-value is less than the significance level, it suggests strong evidence to reject the null hypothesis, indicating that the coin might not be fair.
Takeaways
- 🎲 Hypothesis testing involves evaluating the outcome of an experiment under the assumption that a null hypothesis is true.
- 🧐 The null hypothesis is a default position that there's no effect or no difference, which is what we assume to be true until evidence suggests otherwise.
- 🤔 An alternative hypothesis is what you might believe to be true if the null hypothesis is proven false.
- 👍 A fair coin has a 50/50 chance of landing on heads or tails, which serves as the basis for the null hypothesis in the coin-tossing experiment.
- 🔄 Each coin toss is an independent event with a 50% chance of heads, given the assumption of a fair coin.
- 📉 The probability of multiple heads in a row decreases with each additional toss if the coin is fair, following a binomial distribution.
- 🤏 The significance level (alpha) is the threshold for deciding when the observed results are unlikely enough under the null hypothesis to reject it.
- 📉 A p-value is the probability of observing a result as extreme as, or more extreme than, the one calculated if the null hypothesis is true.
- ✋ If the p-value is less than the significance level, it suggests that the null hypothesis can be rejected in favor of the alternative hypothesis.
- 💰 The speaker uses a betting scenario to illustrate the concept of confidence in the fairness of the coin and the decision to reject the null hypothesis.
- 🔮 The concept of 'unusual' in the context of a p-value is subjective and depends on the significance level set for the experiment.
- 🚫 Hypothesis testing does not always reveal the true state of the world (e.g., the fairness of the coin); it provides a statistical method to make decisions based on evidence.
Q & A
What is a p-value in the context of hypothesis testing?
-A p-value is the probability of observing a result as extreme as, or more extreme than, the one calculated from my test data, under the assumption that the null hypothesis is true.
What is the null hypothesis?
-The null hypothesis is a statement that there is no effect or no relationship between the variables being studied, and it is what we assume to be true at the start of an experiment.
What is the alternative hypothesis?
-The alternative hypothesis is a statement that contradicts the null hypothesis, suggesting that there is an effect or a relationship between the variables.
Why do we perform hypothesis tests?
-Hypothesis tests are performed to determine whether the results of an experiment or study are likely to have occurred by chance, or if they provide evidence of a genuine effect or relationship.
What is the significance level (alpha) in a hypothesis test?
-The significance level (alpha) is a threshold probability that determines when we reject the null hypothesis. It is the probability of rejecting the null hypothesis when it is actually true.
How is the significance level (alpha) determined in a hypothesis test?
-The significance level is typically set by the researcher before conducting the test, based on how much risk they are willing to take of making a Type I error (falsely rejecting the null hypothesis).
What does it mean to reject the null hypothesis?
-Rejecting the null hypothesis means that the evidence from the data provides enough support to suggest that the alternative hypothesis is true, and that the observed results would be unlikely to occur if the null hypothesis were correct.
What is the probability of getting six heads in a row when flipping a fair coin?
-The probability of getting six heads in a row when flipping a fair coin is (1/2)^6, which is approximately 1.5625% or 0.0156 in decimal form.
What is the role of the assumption of a fair coin in the experiment?
-The assumption of a fair coin is a starting point for the experiment. It allows us to calculate the expected probabilities of outcomes under the null hypothesis that the coin is fair, which we can then compare to the actual results of the coin flips.
Why is the probability calculation different if the coin is not fair?
-If the coin is not fair, the probabilities of landing on heads or tails are not equal to 50% each. Without knowing the actual probabilities, we cannot accurately calculate the chances of a particular sequence of outcomes.
What is the concept of 'confidence' in the context of hypothesis testing?
-In hypothesis testing, 'confidence' refers to the level of certainty we have that the decision to reject or fail to reject the null hypothesis is correct. A higher confidence level means we are more certain about our decision.
Why is it important to not look at the coin after the hypothesis test in the script?
-Looking at the coin after the hypothesis test would introduce a form of 'peeking' or post hoc information that could bias the results. The integrity of the hypothesis test lies in making a decision based solely on the data collected without any additional information.
Outlines
🎲 Understanding P-Values and Hypothesis Testing
The first paragraph introduces the concept of p-values and hypothesis testing through a conversation between two individuals. They discuss the idea of a fair coin and the probability of getting heads or tails, which leads to the concept of a null hypothesis. The null hypothesis is assumed to be true until evidence suggests otherwise. The conversation uses a coin-tossing experiment to illustrate how repeated outcomes can provide more information about the fairness of the coin. The significance of the p-value is explained as the probability of observing an unusual result if the null hypothesis were true. The discussion concludes with the idea of setting a significance level (alpha) to decide when the evidence is strong enough to reject the null hypothesis.
🔍 Defining P-Value and Significance Level
The second paragraph delves deeper into the definition of the p-value and the concept of a significance level (alpha). The p-value is clarified as the probability of obtaining a result as unusual or more unusual than what was observed, assuming the null hypothesis is true. The conversation touches on the discomfort with the term 'unusual' and promises a more detailed explanation in a subsequent video. The discussion emphasizes that the p-value is used to compare with the significance level to determine whether to reject the null hypothesis. The speaker also humorously refuses to reveal the coin used in the experiment, highlighting the abstract nature of hypothesis testing where the actual coin isn't revealed until after the analysis is complete.
Mindmap
Keywords
💡p-value
💡hypothesis test
💡null hypothesis
💡alternative hypothesis
💡significance level (alpha)
💡confidence level
💡coin toss
💡probability
💡unusual result
💡evidence
💡rejection of null hypothesis
Highlights
A p-value is a statistical measure used to evaluate the outcome of a hypothesis test.
A hypothesis test involves comparing a null hypothesis (what is believed to be true) against an alternative hypothesis.
In the example, the null hypothesis is that a coin is fair, meaning it has an equal chance of landing on heads or tails.
The concept of a fair coin is used to calculate the probability of different outcomes in repeated tosses.
The probability of getting a certain number of heads in a row can be calculated if the coin is assumed to be fair.
As more evidence (coin tosses) is gathered, the likelihood of the null hypothesis being true can be evaluated.
If the observed outcome is highly unlikely under the null hypothesis, it provides evidence against the null hypothesis.
A significance level (alpha) is set to determine the threshold for rejecting the null hypothesis.
The significance level represents the probability of rejecting the null hypothesis when it is actually true.
In the example, getting 6 heads in a row is considered strong enough evidence to reject the null hypothesis at a 3% significance level.
The p-value is the probability of observing a result as extreme as, or more extreme than, the actual outcome if the null hypothesis is true.
If the p-value is less than the significance level, it suggests that the null hypothesis should be rejected.
The example demonstrates the concept of hypothesis testing using a coin toss experiment.
The experiment shows how the p-value and significance level can be used to make a statistical decision.
The process of hypothesis testing helps to determine whether the observed data supports the null hypothesis or the alternative hypothesis.
It's important to note that rejecting the null hypothesis does not definitively prove the alternative hypothesis is true.
The concept of hypothesis testing and p-values can be applied to a wide range of scientific and research scenarios.
Transcripts
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