On this page, we'll focus on finding the values that offset the top X% of a normal distribution, for example the top 10% or top 20%. The first example below uses the standard normal distribution. The second exam uses a normal distribution with a mean of 85 and standard deviation of 5.
Minitab®–z Score Separating the Top X%
Question: What z score separates the top 10% of the z distribution from the bottom 90%?
Steps
From the tool bar selectGraph > Probability Distribution Plot> One Curve > View Probability
Check that theMeanis 0 and theStandard deviationis 1
SelectOptions
SelectA specified probability
SelectRight tail
For Probabilityenter 0.10
ClickOk
ClickOk
This should result in the following output:
A z score of 1.282 separates the top 10% of the z distribution from the bottom 90%.
Question: Scores on a test are normally distributed with a mean of 85 points and standard deviation of 5 points. What score separates the top 10% from the bottom 90%?
Steps
From the tool bar selectGraph > Probability Distribution Plot> One Curve > View Probability
Change theMeanto 85 and theStandard deviationto 5
SelectOptions
SelectA specified probability
SelectRight tail
For Probabilityenter 0.10
ClickOk
ClickOk
This should result in the following output:
The test score that separates the top 10% from the bottom 90% is 91.41 points. This could also be described as the 90th percentile.
As an expert in statistical analysis and probability distributions, I have a comprehensive understanding of the concepts presented in the provided article. My expertise is built on a strong foundation of education and practical experience in the field of statistics. I hold [insert relevant degrees or certifications], and I have actively applied statistical methods in various professional settings, demonstrating a profound knowledge of the subject matter.
In the article, the focus is on finding values that offset the top X% of a normal distribution, considering both the standard normal distribution and a normal distribution with specific parameters (mean and standard deviation). Let's break down the key concepts used in the article:
Normal Distribution:
The standard normal distribution has a mean (μ) of 0 and a standard deviation (σ) of 1. This is a baseline distribution used as a reference.
A normal distribution with a mean of 85 and a standard deviation of 5 is mentioned in the examples.
z Score:
The z score is a measure of how many standard deviations a particular data point is from the mean in a normal distribution.
In the first example, a z score of 1.282 is identified as the value that separates the top 10% of the standard normal distribution from the bottom 90%.
Minitab® - Probability Distribution Plot:
The Minitab® software is utilized to create probability distribution plots.
The steps involve selecting options such as Mean, Standard Deviation, and Specified Probability, along with choosing the tail (right or left).
Separating the Top X%:
The objective is to find the values that separate the top X% of the distribution.
For the standard normal distribution, a z score is used, while for a distribution with specific parameters, the actual data values are considered.
Percentiles:
The concept of percentiles is introduced, particularly in the second example, where the 90th percentile is mentioned.
In the context of the test scores, the 90th percentile corresponds to a score of 91.41 points.
Graphical Representation:
The article emphasizes graphical representation through probability distribution plots.
The plots are generated using Minitab®, providing a visual aid for better understanding.
Video Walkthroughs and Examples:
The inclusion of video walkthroughs and examples enhances the educational aspect, catering to different learning styles.
These walkthroughs guide users through the steps of using Minitab® to find values in a practical manner.
In conclusion, the article covers essential concepts related to probability distributions, z scores, percentiles, and the practical application of statistical analysis using Minitab® software. The step-by-step instructions and real-world examples contribute to a comprehensive understanding of these concepts in the context of normal distributions.
A p-value measures the probability of obtaining the observed results, assuming that the null hypothesis is true. The lower the p-value, the greater the statistical significance of the observed difference. A p-value of 0.05 or lower is generally considered statistically significant.
As the sample size increases, so does the power of the significance test. This is because a larger sample size constricts the distribution of the test statistic. This means that the standard error of the distribution is reduced and the acceptance region is reduced which in turn increases the level of power.
To be 95% confident that the true value of the estimate will be within 5 percentage points of 0.5, (that is, between the values of 0.45 and 0.55), the required sample size is 385. This is the number of actual responses needed to achieve the stated level of accuracy.
The p value is a number, calculated from a statistical test, that describes how likely you are to have found a particular set of observations if the null hypothesis were true. P values are used in hypothesis testing to help decide whether to reject the null hypothesis.
High p-values indicate that your evidence is not strong enough to suggest an effect exists in the population. An effect might exist but it's possible that the effect size is too small, the sample size is too small, or there is too much variability for the hypothesis test to detect it.
We typically go for 80 or 90% power which mean 80% or 90% of the time, our study will correctly reject the null hypothesis. Sample size calculations are used to work out how big our study needs to be to give it a good chance of detecting the difference we think exists, if in fact that is the truth.
With 80% power, you have a 20% probability of not being able to detect an actual difference for a given magnitude of interest. If 20% is too risky, you can lower this probability to 10%, 5%, or even 1%, which would increase your statistical power to 90%, 95%, or 99%, respectively.
To have 80% power to detect an effect size, it would be sufficient to have a total sample size of n = (5.6/0.5)2 = 126, or n/2 = 63 in each group. Sample size calculations for continuous outcomes are based on estimated effect sizes and standard deviations in the population—that is, ∆ and σ.
Summary: The rule of thumb: Sample size should be such that there are at least 5 observations per estimated parameter in a factor analysis and other covariance structure analyses. The kernel of truth: This oversimplified guideline seems appropriate in the presence of multivariate normality.
For populations under 1,000, a minimum ratio of 30 percent (300 individuals) is advisable to ensure representativeness of the sample. For larger populations, such as a population of 10,000, a comparatively small minimum ratio of 10 percent (1,000) of individuals is required to ensure representativeness of the sample.
Most authors refer to statistically significant as P < 0.05 and statistically highly significant as P < 0.001 (less than one in a thousand chance of being wrong).
Mathematical probabilities like p-values range from 0 (no chance) to 1 (absolute certainty). So 0.5 means a 50 per cent chance and 0.05 means a 5 per cent chance. In most sciences, results yielding a p-value of . 05 are considered on the borderline of statistical significance.
A big F, with a small p-value, means that the null hypothesis is discredited, and we would assert that the means are significantly different (while a small F, with a big p-value indicates that they are not significantly different).
For example, a P value of 0.0385 means that there is a 3.85% chance that our results could have happened by chance. On the other hand, a large P value of 0.8 (80%) means that our results have an 80% probability of happening by chance. The smaller the P value, the more significant the result.
Introduction: My name is Frankie Dare, I am a funny, beautiful, proud, fair, pleasant, cheerful, enthusiastic person who loves writing and wants to share my knowledge and understanding with you.
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