P Values, z Scores, Alpha, Critical Values

statisticsfun
7 Apr 201505:37
EducationalLearning
32 Likes 10 Comments

TLDRThis video script offers an insightful discussion on statistical concepts that often perplex students, namely P values, alpha levels, z-scores, and critical values. The presenter begins by illustrating critical values in the context of a 95% confidence level, explaining how they define the rejection region on a bell curve. The script then delves into the process of comparing calculated z-scores to critical values to determine the validity of the null hypothesis. The concept of alpha is introduced as the area in the tails of the distribution, which is crucial for two-tailed tests. The script also clarifies the role of P values in assessing the significance of results, demonstrating how to calculate or look up exact P values using a standardized table. The presenter emphasizes the importance of understanding these statistical tools for both academic and professional research, encouraging viewers to explore related videos for a deeper comprehension of the subject.

Takeaways
  • ๐Ÿ“Š **Critical Values**: These mark the boundary of the rejection region in a hypothesis test, determining whether to reject the null hypothesis based on the test statistic.
  • ๐ŸŽฏ **95% Confidence Level**: The video uses a 95% confidence level, which implies a 5% chance of rejecting the null hypothesis when it is true (Type I error).
  • ๐ŸŸฅ **Rejection Region**: The area beyond the critical value where the null hypothesis is rejected. In a two-tailed test, this area is split between the two tails of the distribution.
  • ๐Ÿ”ข **Z-Score Comparison**: The calculated z-score from a test is compared against the critical value to decide if the null hypothesis should be rejected.
  • ๐ŸŒฟ **Alpha (ฮฑ)**: Represents the total area in the tails of the distribution (5% in this case), which is the probability of a Type I error.
  • ๐Ÿงฎ **Calculating Alpha**: Alpha is split evenly between the two tails of the distribution, hence ฮฑ/2 represents the critical value for each tail.
  • ๐Ÿ“‰ **P-Value**: Indicates the probability of observing a result as extreme as, or more extreme than, the one calculated from the data, assuming the null hypothesis is true.
  • ๐Ÿ“ˆ **Z-Score and P-Value**: A z-score of 2.6 leads to a p-value less than 0.025, showing a significant result that falls in the rejection region.
  • ๐Ÿ” **Looking Up P-Values**: The exact p-value can be found using a standard normal distribution table or calculated precisely.
  • ๐Ÿ“š **Statistical Tables**: Reference tables provide z-scores and their corresponding p-values, which are crucial for hypothesis testing.
  • ๐Ÿ“ **Reporting P-Values**: It's acceptable to report a p-value as either the exact value (e.g., 0.0047) or as less than a threshold (e.g., <0.025), depending on requirements.
  • ๐Ÿ“Œ **Submission Standards**: When submitting to a journal, adhere to the specific guidelines regarding how to report p-values and statistical significance.
Q & A
  • What are the statistical concepts discussed in the video?

    -The video discusses P values, alpha, z-scores, and critical values, focusing on their commonalities and differences.

  • What is the significance of a critical value in hypothesis testing?

    -A critical value is the boundary of the rejection region in a hypothesis test. If the test statistic falls into the rejection region, the null hypothesis is rejected.

  • What is the confidence level discussed in the video?

    -The video discusses a 95% confidence level, which means that 95% of the observations fall within the confidence interval.

  • How is the rejection region determined in a hypothesis test?

    -The rejection region is determined by the critical value, which marks the boundary. If the test statistic is beyond this boundary, the null hypothesis is rejected.

  • What does the z-score represent in the context of the video?

    -In the video, the z-score represents the test result of a statistical experiment. It is compared to the critical value to decide whether to reject the null hypothesis.

  • How is alpha related to the tails of the distribution in a two-tailed test?

    -Alpha is the total area in the tails of the distribution. In a two-tailed test, alpha is divided by two and placed in both tails, representing the probability of a type I error.

  • What does the term 'P-value' signify in statistics?

    -The P-value is the probability of obtaining a result as extreme as, or more extreme than, the observed result under the assumption that the null hypothesis is true. It indicates the significance of the result.

  • How can one find the exact P-value for a given z-score?

    -The exact P-value can be found by looking up the z-score in a standard normal distribution table or by calculating it using statistical software or formulas.

  • What is the relationship between the P-value and the area under the curve in a standard normal distribution?

    -The P-value corresponds to the area under the curve of the standard normal distribution to the right of the observed z-score in a one-tailed test, or split between both tails in a two-tailed test.

  • Why is it important to understand the concept of a 'two-tailed test'?

    -A two-tailed test is important because it considers the possibility of extreme results in both directions from the mean, not just one. This makes it more sensitive to detecting significant differences or effects.

  • What does the video suggest for further learning?

    -The video suggests that viewers watch related videos for a deeper understanding of the concepts discussed, such as how to calculate P-values and work with z-scores.

  • How can one enhance their understanding of statistical concepts like P-values, alpha, and z-scores?

    -One can enhance their understanding by practicing with real data, referring to statistical tables, using software tools, and engaging with educational resources like the videos mentioned in the script.

Outlines
00:00
๐Ÿ“Š Understanding Critical Values, Z-Scores, and P-values

This paragraph introduces the concepts of critical values, Z-scores, and P-values, which are often confusing for students. The speaker uses a bell curve to illustrate these concepts and focuses on a 95% confidence level. Critical values are explained as the boundary of the rejection region in a statistical test. The video shows how a Z-score of 2.6 falls within the rejection region, leading to the rejection of the null hypothesis. Alpha is described as the area in the tails of the distribution, which in this example is 0.025 for a two-tailed test. The P-value is introduced as a measure of significance, with the speaker calculating it to be less than 0.025 for the Z-score of 2.6. The exact P-value is further explored, with the speaker demonstrating how to find it in a normalized table or calculate it, resulting in an exact P-value of 0.0047.

05:01
๐Ÿ“ˆ P-value Reporting and Submission Guidelines

The second paragraph discusses how to report P-values and the flexibility in presenting them. It is mentioned that one can report the P-value as either equal to 0.0047 or less than 0.0250, and both are considered correct. The choice depends on the requirements set by an academic professor or a journal to which one might be submitting research. The speaker also encourages viewers to watch related videos for a deeper understanding and to share the knowledge on social media platforms. The paragraph ends with a call to action for viewers to subscribe and engage with the content.

Mindmap
Keywords
๐Ÿ’กCritical Values
Critical values are the thresholds used in statistical hypothesis testing that determine the rejection or non-rejection of a null hypothesis. In the context of the video, a critical value of 1.96 is used for a 95% confidence level, marking the boundary of the rejection region. If a calculated test statistic, such as a z-score, exceeds this critical value, the null hypothesis is rejected, indicating a statistically significant result.
๐Ÿ’กConfidence Level
A confidence level, often expressed as a percentage, represents the degree of certainty that the true value of a population parameter lies within a confidence interval. The video discusses a 95% confidence level, which means that if the experiment were repeated many times, the rejection region would encompass the true parameter value 95% of the time.
๐Ÿ’กRejection Region
The rejection region is the area in a statistical distribution where the null hypothesis is rejected if the test statistic falls within it. In the video, the red shaded area represents the rejection region for a 95% confidence level, and it is where the test statistic would lead to the conclusion that the null hypothesis is not supported by the data.
๐Ÿ’กZ-Score
A z-score is a measure of how many standard deviations an element is from the mean in a normal distribution. In the video, the speaker calculates a z-score of 2.6 during an experiment and compares it to the critical value to determine if the null hypothesis should be rejected. The z-score is a key component in hypothesis testing as it quantifies the distance of an observed value from the expected value under the null hypothesis.
๐Ÿ’กNull Hypothesis
The null hypothesis is a statement that there is no significant difference or effect in a study, which is tested against an alternative hypothesis. In the video, the null hypothesis is rejected when the calculated z-score is found to be in the rejection region, indicating that the observed result is unlikely to have occurred by chance alone.
๐Ÿ’กAlpha
Alpha (ฮฑ) is the probability of rejecting the null hypothesis when it is true. It is also the significance level of a hypothesis test. In the video, alpha is set at 0.05, which means there is a 5% chance of a Type I error (false positive). The alpha value is split between the two tails of the distribution, with ฮฑ/2 in each tail for a two-tailed test.
๐Ÿ’กP-Value
The p-value is the probability of obtaining results at least as extreme as the observed results under the assumption that the null hypothesis is true. In the video, the p-value is calculated to be less than 0.025, indicating a low probability that the observed z-score of 2.6 could have occurred by chance, thus suggesting that the null hypothesis is likely false.
๐Ÿ’กTwo-Tailed Test
A two-tailed test is a type of statistical test where the rejection region is divided equally between the two tails of a statistical distribution. The video explains that in a two-tailed test, the alpha level is split between the two tails, which is why ฮฑ/2 is used in the calculations. This type of test is used when the direction of the effect is unknown or when the hypothesis is non-directional.
๐Ÿ’กBell Curve
The bell curve, also known as the normal distribution, is a type of continuous probability distribution that is symmetric about the mean and is characterized by a bell-shaped graph. In the video, the bell curve is used to illustrate the distribution of z-scores and to identify the critical values and rejection regions for a given confidence level.
๐Ÿ’กSignificance Level
The significance level is the threshold for deciding whether to reject the null hypothesis. It is often denoted by the Greek letter alpha (ฮฑ) and is the probability of committing a Type I error. In the video, the significance level is set at 0.05, which means that there is a 5% chance of incorrectly rejecting a true null hypothesis.
๐Ÿ’กType I Error
A Type I error occurs when the null hypothesis is incorrectly rejected when it is actually true. It is also known as a false positive. The video mentions that the alpha level represents the probability of making a Type I error, and in the context of the 95% confidence level discussed, there is a 5% chance of making such an error.
๐Ÿ’กNormalized Table
A normalized table, often referred to as a z-table, is a statistical tool that provides the probability associated with a given z-score in a standard normal distribution. In the video, the speaker uses a normalized table to find the p-value corresponding to a z-score of 2.6, which is crucial for determining the significance of the test result.
Highlights

The video discusses P values, alpha, z-scores, and critical values, which are common points of confusion for students.

A 95% confidence level is introduced, with 1.96 as the critical value for the rejection region.

Critical values act as the boundary of the rejection region in hypothesis testing.

The concept of a z-score is explained, with an example of 2.6 being in the rejection region.

Alpha is defined as the area in the tails of the distribution, calculated as 0.025 for a two-tailed test at a 95% confidence level.

The green area represents 95% of observations, with the remaining 5% split between the two tails.

The significance of a result is determined by the p-value, which is the probability of obtaining the result by random chance.

The p-value for a z-score of 2.6 is less than 0.025, indicating a significant result.

The exact p-value can be calculated or looked up in a normalized table of z-scores and probabilities.

The video provides a link to another resource that explains how to calculate or find p-values in more detail.

The concept of alpha divided by two is explained in the context of two-tailed tests and its use in equations like confidence intervals.

The video emphasizes that both stating 'P is equal to 0.0047' and 'P is less than 0.025' are correct ways to express p-values.

The importance of understanding the requirements of professors or journals when reporting p-values is highlighted.

The video encourages viewers to watch related videos for a deeper understanding of statistical concepts.

The presenter shares links to related videos for further learning and understanding.

The video concludes with a call to action for viewers to share the knowledge, like, subscribe, and engage with the content.

Transcripts
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