Introduction to power in significance tests | AP Statistics | Khan Academy

Khan Academy
31 Jan 201809:45
EducationalLearning
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TLDRThe video script delves into the concept of statistical power in significance testing, a topic often introduced in first-year statistics courses. Power is defined as the probability of correctly rejecting a false null hypothesis, which can also be viewed as one minus the probability of committing a Type II error. The script explains that power can be influenced by several factors, including the significance level (alpha), sample size (n), variability within the dataset, and the distance of the true parameter from the null hypothesis. Increasing the alpha level boosts power but also raises the risk of a Type I error. Conversely, a larger sample size or less variability in the data inherently increases power without such trade-offs. The video aims to provide a clear understanding of what power represents and how it can be improved, emphasizing the importance of sample size and the delicate balance between alpha level adjustments and the potential for Type I errors.

Takeaways
  • πŸ“Š **Power Definition**: Power is the probability of correctly rejecting the null hypothesis when it is false.
  • πŸ” **Type II Error Connection**: Power can also be viewed as one minus the probability of making a Type II error, which is not rejecting the null hypothesis when it is false.
  • πŸ“ˆ **Sampling Distributions**: Two sampling distributions are considered: one under the assumption that the null hypothesis is true, and another when it is false.
  • 🎯 **Significance Level**: The significance level (alpha) is the probability of rejecting the null hypothesis when it is true, and it is also related to the power of the test.
  • πŸ”΄ **Increasing Power**: Power can be increased by raising the significance level, which however increases the risk of a Type I error.
  • πŸ”΅ **Sample Size Impact**: Increasing the sample size (n) narrows the sampling distributions and reduces the overlap, thus increasing power.
  • πŸ“Š **Variability Reduction**: Less variability in the dataset, as measured by variance or standard deviation, leads to narrower sampling distributions and higher power.
  • πŸ”— **Effect Size**: The greater the difference between the true parameter value and the null hypothesis value, the higher the power of the test.
  • βš–οΈ **Trade-offs**: There's a trade-off between increasing power and the risk of Type I errors when adjusting the significance level.
  • πŸ”§ **Control Factors**: Researchers can control the sample size and to some extent the significance level to influence power, but other factors like variability and effect size are often outside of their control.
  • βœ… **Best Practice**: Generally, increasing the sample size is a good strategy to increase power without introducing more risk of Type I errors.
Q & A
  • What is the concept of power in the context of significance tests?

    -Power is the probability that you correctly reject the null hypothesis when it is not true. It is the likelihood of avoiding a Type II error, which is not rejecting the null hypothesis when it should be rejected.

  • How is power related to Type II errors?

    -Power is equal to one minus the probability of making a Type II error. It is the probability of correctly rejecting the null hypothesis when it is false.

  • What happens to the sampling distribution if the null hypothesis is true?

    -If the null hypothesis is true, the sampling distribution's center would be at the hypothesized parameter value (mu one), and the distribution's width would be influenced by the sample size.

  • How does the sample size affect the sampling distribution?

    -The sample size affects the width of the sampling distribution. A larger sample size results in a narrower distribution, while a smaller sample size results in a wider distribution.

  • What is the significance level in a significance test?

    -The significance level is the probability of rejecting the null hypothesis when it is true, which is also the probability of making a Type I error.

  • How does increasing the alpha level affect the power of a test?

    -Increasing the alpha level increases the power of a test because it expands the area under the sampling distribution that would lead to the rejection of the null hypothesis. However, it also increases the probability of a Type I error.

  • What is the impact of increasing the sample size on the power of a test?

    -Increasing the sample size narrows both sampling distributions, reducing the overlap between them and thus reducing the probability of a Type II error, which increases the power of the test.

  • How does the variability in the dataset affect the power of a test?

    -Less variability in the dataset results in narrower sampling distributions, which reduces the area representing the probability of a Type II error and increases the power of the test.

  • What is the effect of the true parameter being further from the null hypothesis on the power?

    -If the true parameter is further away from what the null hypothesis states, it increases the power because it increases the separation between the sampling distributions under the null and alternative hypotheses.

  • Why might a researcher be hesitant to increase the significance level to boost power?

    -A researcher might be hesitant because increasing the significance level also increases the probability of a Type I error, which is the incorrect rejection of a true null hypothesis.

  • What is the trade-off involved in deciding the significance level?

    -The trade-off is between increasing power and the risk of a Type I error. A higher significance level increases power but also the chance of incorrectly rejecting a true null hypothesis.

  • Why is increasing the sample size generally considered a good strategy to increase power?

    -Increasing the sample size is beneficial because it narrows the sampling distributions, reducing the chance of a Type II error without increasing the risk of a Type I error.

Outlines
00:00
πŸ“Š Understanding Statistical Power

This paragraph introduces the concept of power in the context of significance tests, which is a fundamental topic in introductory statistics. Power is defined as the probability of correctly rejecting a false null hypothesis. It's portrayed as a conditional probability and is also related to the concept of Type II errors. The instructor emphasizes the importance of understanding what factors can increase or decrease the power of a test, and prepares to illustrate this with sampling distributions under both the null hypothesis and an alternative scenario where the null hypothesis is false.

05:01
🎯 Factors Influencing Statistical Power

The second paragraph delves into the factors that can affect the power of a statistical test. It explains how increasing the significance level (alpha) can boost power but also raises the risk of a Type I error. The paragraph also highlights the impact of sample size on power, noting that a larger sample size narrows the sampling distribution and reduces the chance of a Type II error. Variability in the dataset, represented by variance or standard deviation, is another factor that can influence power, with less variability leading to more distinct sampling distributions. Lastly, the true parameter value's distance from the null hypothesis's claim is mentioned as a factor that can increase power, although this is often not under the researcher's control. The paragraph concludes by reiterating the importance of sample size and the trade-offs involved in adjusting the significance level.

Mindmap
Keywords
πŸ’‘Power
In the context of the video, 'power' refers to the probability of correctly rejecting the null hypothesis when it is false. It is a key concept in significance testing within statistics. The video emphasizes that power is important because it measures the study's ability to detect an effect when there is one. An example from the script is the discussion on how power can be increased by manipulating factors such as sample size or significance level.
πŸ’‘Significance Test
A 'significance test' is a statistical method used to determine whether a result reflects a real effect or is due to random chance. It is central to the video's theme as it sets the stage for discussing the concept of power. The script mentions that power is a concept that might be encountered in a first-year statistics course in the context of significance tests.
πŸ’‘Null Hypothesis
The 'null hypothesis' is a statement that there is no effect or no difference between groups being studied. It is a fundamental concept in the video as power is defined as the probability of correctly rejecting this hypothesis when it is false. The script uses the example of a null hypothesis stating that a population mean equals 'mu one'.
πŸ’‘Alternative Hypothesis
The 'alternative hypothesis' is a statement that contradicts the null hypothesis by suggesting an effect or a difference does exist. It is directly related to the concept of power as the power of a test is the probability of correctly accepting the alternative hypothesis when it is true. In the script, the alternative hypothesis is symbolized as 'H sub a' which posits that the population mean is not equal to 'mu one'.
πŸ’‘Type II Error
A 'Type II error' occurs when the null hypothesis is false, but it is not rejected by the test. It is related to the video's theme as power is defined as one minus the probability of making a Type II error. The script illustrates this by saying that power is the probability of not making this error when the null hypothesis is false.
πŸ’‘Sample Size (n)
The 'sample size' refers to the number of observations or elements in a sample. It is a key factor in determining the power of a test. The video explains that increasing the sample size makes the sampling distributions narrower, thus reducing the chance of a Type II error. The script mentions that increasing 'n' increases power, which is beneficial if feasible.
πŸ’‘Significance Level (alpha)
The 'significance level', often denoted by alpha, is the probability of rejecting the null hypothesis when it is true, which corresponds to the risk of a Type I error. The video discusses that increasing the significance level can increase power, but it also raises the probability of a Type I error. The script uses alpha to illustrate the trade-off between increasing power and the risk of Type I errors.
πŸ’‘Sampling Distribution
A 'sampling distribution' is the probability distribution of a given statistic based on a random sample. It is central to understanding how power is calculated. The video uses the concept to explain how different sampling distributions arise under the assumption of the null hypothesis being true or false. The script describes how these distributions look under different conditions and how they relate to the probability of making Type I and Type II errors.
πŸ’‘Type I Error
A 'Type I error' happens when the null hypothesis is true, but it is incorrectly rejected. The video touches on this concept when discussing the implications of increasing the significance level. The script mentions that the significance level can be viewed as the probability of making a Type I error, which is a key consideration when evaluating the trade-offs in study design.
πŸ’‘Variance
In the context of the video, 'variance' refers to the degree of spread in a set of data points. It is mentioned as a factor that can affect the power of a test. The less variability in the dataset, the narrower the sampling distributions, which increases power. The script indicates that less variability, as measured by variance or standard deviation, can increase the power of a significance test.
πŸ’‘Population Mean (mu)
The 'population mean', often symbolized as 'mu', is the average value of a population. It is a key parameter in the null and alternative hypotheses discussed in the video. The script uses 'mu one' and 'mu two' to illustrate different hypothetical population means under the null and alternative hypotheses, which are central to understanding the concept of power in hypothesis testing.
Highlights

Power in significance tests is the probability of doing the right thing when the null hypothesis is not true.

Power can also be viewed as one minus the probability of making a Type II error.

The significance level is the probability of rejecting the null hypothesis even if it is true, which is also the probability of a Type I error.

Increasing the sample size narrows the sampling distribution and increases the power of the test.

Less variability in the dataset, measured by variance or standard deviation, results in narrower sampling distributions and higher power.

If the true parameter is significantly different from the null hypothesis, it increases the power of the test.

Increasing the alpha level (significance level) increases power but also the probability of a Type I error.

The significance level and sample size are controllable factors that can be adjusted to affect power.

There is a trade-off when increasing the significance level to gain power, as it raises the risk of a Type I error.

Researchers may choose to increase the significance level if they consider a Type II error to be worse.

The sampling distribution under the null hypothesis is centered at the hypothesized parameter value.

The conditional probability of power is calculated given that the null hypothesis is false.

The probability of not rejecting the null hypothesis when it is false defines a Type II error.

The area under the sampling distribution curve beyond the significance level represents the power of the test.

The concept of power is introduced in first-year statistics courses and is important for understanding significance testing.

The width of the sampling distribution is affected by the sample size; larger sample sizes result in narrower distributions.

The probability of making a Type II error is the area under the sampling distribution where the null hypothesis is not rejected when it is false.

Researchers must balance the desire to increase power with the risk of committing a Type I error by adjusting the significance level.

Transcripts
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