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# Confidence Interval And P Value Pdf

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Published: 19.03.2021  In this series of posts, I show how hypothesis tests and confidence intervals work by focusing on concepts and graphs rather than equations and numbers.

## A beginner’s guide to interpreting odds ratios, confidence intervals and p-values

Sign in. Statistical inference is the process of making reasonable guesses about the population's distributio n and parameters given the observed data.

Conducting hypothesis testing and constructing confidence interval are two examples of statistical inference. Hypothesis testing is the process of calculating the probability of observing sample statistics given the null hypothesis is true.

With a similar process, we can calculate the confidence interval with a certain confidence level. A confidence interval is an interval estimation for a population parameter, which is point estimation plus and minus the critical value times sample standard error. This article will discuss the standard procedure of conducting hypothesis testing and estimating confidence intervals in the following different scenarios:.

This article is both served as a tutorial for statistical inference, as well as a cheat-sheet for your reference. The sections below will discuss the procedures in detail, and at the end of the article, I will summarize discussions in two tables for convenience. We need to have assumptions about the underlining distributions when using statistical inference techniques. According to the central limit theory , the distribution of sample means approaches to normal distributions as sample size increases, no matter what the population distribution is.

The test we usually use here is either the student t-test or the Z test. Z test is based on the normal distribution while student t-test is based on a distribution similar to a normal distribution, but with fatter tails. When the sample size is lower than 30 the standard cut-off or the population standard deviation is unknown, we use the student t-test. Otherwise, we use the Z test. For a sample with n observations:.

Hypothesis Testing. Here are the steps for conducting hypothesis testing:. Two tails:. One tail:. The alternative hypothesis H1 is the hypothesis we want to test. Keep in mind that for the student t-test, since the observation is relatively smaller in the sample, we need to specify the degree of freedom to find the right value.

The degree of freedom is defined as n-1, where n is the sample size. The graph below shows the meaning of critical value. Z test is based on a standard normal distribution N 0,1. Critical value at 1. Confidence Interval. The confidence interval is an interval estimate with a certain confidence level for a parameter. It is calculated by the point estimation plus or minus the margin of errors ME :.

The point estimate is just the mean of the sample, and ME is calculated by:. The distribution and the confidence level define the critical values, and the standard error SE is calculated through the sample or population standard deviation. Otherwise, we need to use the z table to calculate the confidence interval:. The z value can be found in the z table. When we observe two samples, we may wonder whether the means from the two samples differ significantly from each other.

If we have reasons to believe that the two samples are uncorrelated with each other, we can test it either use hypothesis testing with the null hypothesis states the means equal to each other, or conduct a confidence interval for the difference of the means and check whether zero is inside the interval. The procedures are quite similar to the one-sample case, with a bit of difference in calculating the test statistics and standard error.

If both samples are not large enough, we can use the t table assuming a t distribution, and calculate the t statistics as follows:. Depending on the practical situation, we can also set the null hypothesis to check whether the difference between the two means are greater than a certain number that is larger than 0, which sometimes is referred as the effect size. A larger effect size makes it easier to reject the null hypothesis since the difference is bigger, thus increases the statistical power.

For more details, you can check out my article here:. The test statistics for the null hypothesis becomes:. The confidence interval for two sample means are used to describe the difference of the two mean. Using the t critical value, we can calculate the confidence interval as follows:.

Note that similar to the discussion above, with different assumptions about the population variance, we can calculate the standard error in the margin of error term differently. In the previous section, we have discussed the situation when the two samples are independent from each other. What about the situation when two samples are correlated with each other in some way?

For example, the two samples come from the same subjects before and after the treatment, or the samples were taken from different people in the same household, etc. For example, if we want to test whether there are treatment effect in the treat group, we can collect samples before and after the treatment:. We need to calculate the difference before and after treatment for each individual, and get the sample of observed difference:.

In such a way, we have transformed a two-samples case into a one-sample case. Following the procedure discussed above, first we need to calculate the mean and standard deviation of the sample of difference:.

We can set up the null hypothesis based on the practical situation. Typical null hypothesis and alternative hypothesis for a two-tailed test are:. The test statistics is calculated as follows:. Depending on the samples, we can choose to conduct t test or z test. We can also construct a confidence interval for the samples differences. We only need the difference mean, and difference standard deviation to construct the interval.

A confidence interval based on the student t distribution is:. Mean measures the central tendency of continuous variables, but it cannot be used in categorical variables. The proportion of category i in a sample with n categories is calculated by:.

Here I will use a simple example to illustrate the process. Rather than following the normal distribution for mean statistical inference, we use the binomial distribution for binary classification proportions. According to the binomial distribution properties, as the sample size gets larger, the binomial distribution approaches a normal distribution. If not, we will use the student t distribution for the inference. One sample proportion calculates the proportion of a category in a sample.

As discussed above, a use case of one-sample proportion is to test whether a coin is unbiased. Hypothesis testing for one-sample proportion follows similar setting up procedures.

Using the coin-tossing example above, if we want to test whether a coin is unbiased, giving a sample of coin tossing results:. It is the same as testing:. Note this is two-tail testing. After that, suppose the sample size is large enough, we can then calculate the z statistics:. The denominator is used to calculate the standard error for this sample deriving from binomial distribution. Following the same procedure described above, we can use the z table or the t table to find the critical values.

By comparing the statistics calculated here, we can decide whether to reject the null hypothesis or not. The confidence interval for proportion follows the same pattern as the statistical inference for mean, which is using the point estimate and margin of errors, except the standard errors are calculated differently here :. Note that the standard error for hypothesis testing is different from the confidence interval.

Two-Samples proportion test compares the proportions in two samples, which is widely used in AB testing. For example, when we compare the conversion rate between the treatment and control group to see whether there exists a significant treatment effect, we need to test whether the difference in the conversion rate is significantly enough.

We can use hypothesis testing to test whether the two proportions difference, or construct a confidence interval for the difference. Based on the two samples we have, we can calculate the two proportions P1 and P2. To test whether the two proportions are not significantly different from each other, meaning that the two samples could be drawn from the same population, the null hypothesis and alternative hypothesis are:.

H0: P1! Note that this is a two-tailed test. For one-tailed test, we can check whether P1 is greater than or less than P2 in the null hypothesis. You can understand it as we are pooling the two samples together, what is the proportion of category i in the pooled sample. If not, we need to calculate the t statistics. The statistics is calculated by:. If both sample sizes are large enough, we can use the critical value z from the z table and calculate the confidence interval as:.

The only difference between the confidence interval and hypothesis testing is the calculation of standard error. Instead of using the pooled proportion, confidence interval uses the standard errors for each sample individually. Here I use two tables to summarize the main takeaways of this article:. Thank you for reading. Here is the list of all my blog posts. Check them out if you are interested!

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Posted on 13th August by Tim Hicks. Students of medicine or from the clinical sciences and professions allied to medicine wanting to enhance their understanding of medical literature they will encounter throughout their careers. The first steps in learning to understand and appreciate evidence-based medicine are daunting to say the least, especially when confronted with the myriad of statistics in any paper. This short tutorial aims to introduce healthcare students to the interpretation of some of the most commonly used statistics for reporting the results of medical research. The scenario for this tutorial is centred around the diagram below, which outlines a fictional parallel two arm randomised controlled trial of a new cholesterol lowering medication against a placebo. An odds ratio is a relative measure of effect, which allows the comparison of the intervention group of a study relative to the comparison or placebo group. So when researchers calculate an odds ratio they do it like this:. PDF | Misinterpretation and abuse of statistical tests, confidence intervals, and statistical power have been decried for decades, yet remain.

## The Ultimate Guide to Hypothesis Testing and Confidence Intervals in Different Scenarios

Misinterpretation and abuse of statistical tests, confidence intervals, and statistical power have been decried for decades, yet remain rampant. A key problem is that there are no interpretations of these concepts that are at once simple, intuitive, correct, and foolproof. Instead, correct use and interpretation of these statistics requires an attention to detail which seems to tax the patience of working scientists. This high cognitive demand has led to an epidemic of shortcut definitions and interpretations that are simply wrong, sometimes disastrously so—and yet these misinterpretations dominate much of the scientific literature. In light of this problem, we provide definitions and a discussion of basic statistics that are more general and critical than typically found in traditional introductory expositions.

Metrics details. Here, we summarise the unresolved debate about p value and its dichotomisation. We present the statement of the American Statistical Association against the misuse of statistical significance as well as the proposals to abandon the use of p value and to reduce the significance threshold from 0.

He also advises organizations on their data and data quality programs. Consider the example of a marketing campaign. This is called a sampling error , something you must contend with in any test that does not include the entire population of interest. Redman notes that there are two main contributors to sampling error: the size of the sample and the variation in the underlying population. Sample size may be intuitive enough. #### Introduction

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