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Hypothesis Testing

paired t-test examples

  • STATISTICS
Paired $t$-test (Dependent Sample) In this tutorial we will discuss some numerical examples on paired t test. Example 1 A new prep class was designed to improve AP statistics test scores. Five students were selected at random. The numbers of correct answers on two practice exams were recorded; one before the class and one after. The data recorded in the table below. We want to test if the numbers of correct answers, on average, are higher after the class.
7 min Read 5

t test p value

  • STATISTICS
p value of the test p-value of the test is the probability that the test statistic under null hypothesis will take on values as extreme as or more extreme than the observed value of test statistic. The smaller p-value of the test indicate strong evidence against the null hypothesis $H_0$. In this tutorial we discuss about how to find the $p$-value of the t-test. The p-value of the $t$-test depends on the direction of the alternative hypothesis.
3 min Read 2

t-test for Mean

  • STATISTICS
One Sample t 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 unknown. One sample t test for mean Let $X_1, X_2, \cdots, X_n$ be a random sample from a normal population with mean $\mu$ and unknown variance $\sigma^2$. Let $\overline{x}=\frac{1}{n} \sum x_i$ be the sample mean and $s^2 =\frac{1}{n-1} \sum (x_i -\overline{x})^2$ be the sample variance.
3 min Read 1

t-test for Mean examples

  • STATISTICS
One Sample $t$-Test for mean In this tutorial we will discuss some numerical examples on one sample t-test for testing population mean when population standard deviation is unknown. This tutorial will show you step by step procedure to solve numerical examples on one sample t-test for population mean. Example 1 The Healthy Food Company claims that its cereal boxes contain, on average, 453 grams of cereal. We suspect that the cereal boxes contain, on average, less than claimed.
5 min Read 3

t-test for two means equal variances

  • STATISTICS
Two sample t test with unknown and equal variances In this tutorial we will discuss two sample t-test for testing difference between two population means when the population variances are unknown but equal. Two sample t test with unknown and equal variances Let $\overline{x}_1$ be the sample mean and $s_1$ be the sample standard deviation of a random sample of size $n_1$ from a population with mean $\mu_1$ and variance $\sigma^2_1$.
3 min Read 2

t-test for two means equal variances examples

  • STATISTICS
Two sample t test for means with unknown but equal variances In this tutorial we will discuss some numerical examples on two sample t test for difference between two population means when the population variances are unknown but equal. Example 1 A high school language course is given in two sections, each using a different teaching method. The first section has $21$ students, and the grades in that section have a mean of $82.
6 min Read 4

t-test for two means unequal variances

  • STATISTICS
Two sample t test for means with unknown and unequal variances In this tutorial we will discuss two sample t test for testing difference between two population means when the population variances are unknown and unequal. Two sample t test for means with unknown and unequal variances Let $\overline{x}_1$ be the sample mean and $s_1$ be the sample standard deviation of a random sample of size $n_1$ from a population with mean $\mu_1$ and variance $\sigma^2_1$.
3 min Read 2

t-test for two means unequal variances examples

  • STATISTICS
Two Sample $t$-Test (Independent Sample with Unequal Variances) In this tutorial we will discuss some numerical examples on two sample t test for difference between two population means when the population variances are unknown and unequal. Example 1 A statistician claims that the average score on logical reasoning test taken by students who major in Physics is less than that of students who major in English. The result of the exams, given to 22 Physics students and 33 English students, is shown here.
6 min Read 4

Z Test p value

  • STATISTICS
p value of the test p-value of the test is the probability that the test statistic under null hypothesis will take on values as extreme as or more extreme than the observed value of test statistic. The smaller p-value of the test indicate strong evidence against the null hypothesis $H_0$. In this tutorial we discuss about how to find the $p$-value of the Z-test. The p-value of the $Z$-test depends on the direction of the alternative hypothesis.
3 min Read 1

Z-test for Mean Examples

  • STATISTICS
Z-test for testing population mean In this tutorial we will discuss some numerical examples on one sample z test for population mean. Example 1 The average cost of tuition and room and board at a small private liberal arts college is reported to be $8,500 per term, but a financial administrator believes that the average cost is lower. A study conducted using 350 small liberal arts colleges showed that the average cost per term is $8,445.
5 min Read 3

Z-test for Proportion

  • STATISTICS
$Z$-Test for 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 of successes. Let $p$ be the population proportion of successes and $q = 1- p$ be the population proportion of failures. Assumptions Assumptions for testing a proportion are as follows: a. The sample is a random sample.
3 min Read 1