Z test for Mean

One sample Z-test for mean In this tutorial we will explain the six steps approach used in hypothesis testing to test hypothesis about the population mean when the population standard deviation is known. One Sample Z-Test For Population Mean Let $X_1, X_2, \cdots, X_n$ be a random sample from a normal population with mean $\mu$ …

Z-Test Calculator for Proportion with Examples

$Z$-Test Calculator for single proportion In this tutorial we will discuss $z$-test calculator for testing single proportion with some numerical examples on $Z$-test for testing single population proportion. $Z$-test Calculator for proportion The $Z$-test calculator for testing population proportion makes it easy to calculate the test statistic, $Z$ critical value and the $p$-value given the …

Z-Test for Proportion

Z-Test for Proportion In this tutorial we will discuss about the step by step procedure of one sample $Z$-test for testing population proportion. Let $X$ be the observed number of individuals possessing certain attributes (say, number of successes) in a random sample of size $n$ from a large population, then $\hat{p}=\frac{X}{n}$ be the observed proportion …

Confidence Interval for Variance Examples

Confidence Interval for Variance with Examples In this tutorial we will discuss some numerical examples to understand how to construct a confidence interval for population variance or population standard deviation. Example 1 The mean replacement time for a random sample of 12 microwaves is 8.6 years with a standard deviation of 3.6 years. Construct a …

Confidence Interval for mean sigma unknown

Introduction Confidence interval can be used to estimate the population parameter with the help of an interval with some degree of confidence. One such a parameter that can be estimated is a population mean. In this article we will discuss step by step procedure to construct a confidence interval for population mean when the population …

Confidence Interval for mean sigma unknown examples

This tutorial covers examples on confidence interval for the population mean when the population standard deviation is unknown. Example 1 A new brand of laptop battery is produced by a company. The company claims that the battery will last for an extended period of time before a recharge is necessary. A sample of 40 batteries …

Confidence Interval for mean | Statistics and Probability

Introduction Confidence interval can be used to estimate the population parameter with the help of an interval with some degree of confidence. One such a parameter that can be estimated is a population mean. In this article we will discuss about the derivation and step by step procedure to construct a confidence interval for population …

Confidence Interval Calculator for mean examples

Confidence Interval for Mean Use this calculator to compute the confidence interval for population mean when the population standard deviation is known. Confidence Interval Calculator for mean Sample Size ($n$) Sample Mean ($\overline{x}$) Population Standard Deviation ($\sigma$) Confidence Level ($1-\alpha$) Select90%95%98%99% Calculate Results Standard Error of Mean: Z Critical Value : ($Z$) Margin of Error: …

Bonferroni Inequality

Bonferroni’s Inequality For $n$ events $A_1,A_2,\cdots, A_n$ $$$$\label{bof} P\big(\cap_{i=1}^n A_i\big)\geq \sum_{i=1}^n P(A_i) -(n-1).$$$$ Proof For $n =2$, $$\begin{equation*} P(A_1\cup A_2)= P(A_1)+P(A_2) -P(A_1\cap A_2) \end{equation*}$$ But $P(A_1\cup A_2)\leq 1$. Using this in above equation, we have  \begin{eqnarray*} & &P(A_1\cup A_2)= P(A_1)+P(A_2) -P(A_1\cap A_2)\leq 1 \\ \implies & & P(A_1\cap A_2) …