introduction to hypothesis testing

Hypothesis testing is a statistical method used to determine if a certain hypothesis about a population is
supported by the available data.

In hypothesis testing, we start with a null hypothesis, which represents the status quo or the default
assumption. The null hypothesis usually states that there is no significant difference or relationship
between two or more groups or variables. We then collect data and use statistical tests to evaluate the
evidence against the null hypothesis. If the evidence strongly suggests that the null hypothesis is false,
we reject it and accept an alternative hypothesis, which represents a new claim or theory.

There are two types of errors that can occur in hypothesis testing. A type I error occurs when we reject a
true null hypothesis, while a type II error occurs when we fail to reject a false null hypothesis. The
probability of making a type I error is denoted by alpha (α), while the probability of making a type II
error is denoted by beta (β). The power of a statistical test is the probability of correctly rejecting a false
null hypothesis, and is equal to 1-β.

Hypothesis testing is widely used in various fields, including science, engineering, social sciences,
business, and healthcare, to draw conclusions from data and make informed decisions based on
evidence. It is a powerful tool for testing hypotheses, making predictions, and understanding the
relationships between variables.

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state null and alternative hypothesis

In hypothesis testing, the null hypothesis (H0) represents the default assumption or status quo, and the
alternative hypothesis (H1 or Ha) represents a new claim or theory that we want to test.

The null hypothesis typically states that there is no significant difference or relationship between two or
more groups or variables. For example, in a medical study, the null hypothesis might be that a new drug
has no effect on a particular health outcome, or that there is no difference in the mean weight of two
populations. The null hypothesis is often denoted by the symbol H0.

The alternative hypothesis, on the other hand, is a statement that contradicts the null hypothesis and
asserts that there is a significant difference or relationship between the groups or variables being
studied. For example, in the medical study, the alternative hypothesis might be that the new drug is
effective in treating the health outcome, or that there is a difference in the mean weight of two
populations. The alternative hypothesis is often denoted by the symbol H1 or Ha.

It is important to clearly define the null and alternative hypotheses before conducting any statistical
tests, as they form the basis for evaluating the evidence against the null hypothesis and making
conclusions about the data.

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tests of statistical hypotheses

Tests of statistical hypotheses are methods used to determine whether a given hypothesis about a
population is supported by the available data. These tests involve comparing the observed data to what
we would expect if the null hypothesis were true, and determining the likelihood of observing such data
if the null hypothesis were true.

There are several commonly used tests of statistical hypotheses, including:

Z-Test: This test is used to determine whether the mean of a sample is significantly different from a
known population mean. It is often used when the sample size is large and the population standard
deviation is known.

t-Test: This test is used to determine whether the mean of a sample is significantly different from a
hypothesized population mean. It is often used when the sample size is small, or when the population
standard deviation is unknown.

Chi-Square Test: This test is used to determine whether there is a significant difference between the
observed frequencies of a categorical variable and the expected frequencies under the null hypothesis. It
is often used to test for independence or homogeneity in contingency tables.

ANOVA (Analysis of Variance): This test is used to determine whether there is a significant difference in
the means of two or more groups. It is often used when comparing means across multiple treatments or
groups.

Regression Analysis: This test is used to determine whether there is a significant relationship between
two or more variables. It is often used to model the relationship between a dependent variable and one
or more independent variables.

In order to perform these tests, we must first specify the null and alternative hypotheses, and choose an
appropriate significance level (alpha) to determine the level of evidence required to reject the null
hypothesis. The p-value, which represents the probability of obtaining a result as extreme or more
extreme than the observed data under the null hypothesis, is then calculated and compared to the
significance level to make a conclusion about the hypothesis.

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hypothesis all formuls

There are several common formulas used in hypothesis testing, depending on the specific test being
conducted. Here are some of the most commonly used formulas:

Z-Test:

Z = (x̄ - μ) / (σ / √n)
Where:

x̄ = sample mean

μ = hypothesized population mean

σ = population standard deviation

n = sample size

t-Test:

t = (x̄ - μ) / (s / √n)

Where:

x̄ = sample mean

μ = hypothesized population mean

s = sample standard deviation

n = sample size

Chi-Square Test:

χ^2 = Σ (O - E)^2 / E

Where:

O = observed frequency

E = expected frequency under the null hypothesis

ANOVA:

F = (SSB / (k - 1)) / (SSW / (n - k))

SSB = sum of squares between groups

SSW = sum of squares within groups

k = number of groups

n = total sample size

Regression Analysis:

y = β0 + β1x1 + β2x2 + ... + βpxp + ε

Where:

y = dependent variable

β0 = intercept

β1, β2, ..., βp = coefficients of independent variables

x1, x2, ..., xp = independent variables

ε = error term

These formulas are used to calculate test statistics and p-values, which are then compared to a pre-
determined significance level (alpha) to determine whether to reject or fail to reject the null hypothesis.

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5 math regarding hypotheses

Here are five key mathematical concepts and calculations related to hypothesis testing:
Sample Mean: The sample mean (x)̄ is a measure of central tendency and is calculated by taking the sum
of all the observations in a sample and dividing by the sample size (n).

Standard Deviation: The standard deviation (s) is a measure of the variability or spread of a sample and is
calculated by taking the square root of the sum of the squared differences between each observation
and the sample mean, divided by the sample size minus one (n-1).

Test Statistic: A test statistic is a value calculated from the sample data that is used to determine whether
to reject or fail to reject the null hypothesis. The specific test statistic used will depend on the type of
test being conducted.

Degrees of Freedom: Degrees of freedom (df) refers to the number of independent observations
available for a particular calculation. For example, in a t-test, the degrees of freedom are equal to n-1,
where n is the sample size.

P-Value: The p-value is a probability that measures the strength of the evidence against the null
hypothesis. It represents the probability of observing the sample data, or more extreme data, if the null
hypothesis were true. A p-value below the chosen significance level (alpha) indicates that the evidence is
strong enough to reject the null hypothesis. The specific calculation of the p-value will depend on the
type of test being conducted.

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5 math example regarding hypotheses

Here are five mathematical examples related to hypothesis testing:

Sample Mean: Suppose we have a sample of 20 students and we want to calculate the sample mean of
their test scores. The scores are: 78, 82, 84, 75, 92, 85, 89, 72, 87, 79, 81, 86, 83, 80, 88, 91, 76, 85, 89,
77. The sample mean is calculated by summing the scores and dividing by the sample size:
(78+82+84+75+92+85+89+72+87+79+81+86+83+80+88+91+76+85+89+77) / 20 = 83.1.

deviation to get a measure of the variability in the test scores. The formula is: s = √Σ(x - x)^2
Standard Deviation: Using the same sample of 20 students, we can calculate the sample standard
̄ / (n - 1),
where x is the score and x̄ is the sample mean. Using a calculator or spreadsheet, we get s = 5.95.

use a one-sample t-test and calculate the test statistic using the formula: t = (x̄ - μ) / (s / √n), where x̄ is
Test Statistic: Suppose we want to test the hypothesis that the population mean test score is 85. We can

the sample size. Using the values from the previous examples, we get t = (83.1 - 85) / (5.95 / √20) = -
the sample mean, μ is the hypothesized population mean, s is the sample standard deviation, and n is

1.69.

Degrees of Freedom: Continuing with the previous example, we can calculate the degrees of freedom for
the t-test, which is n-1 = 20-1 = 19.

P-Value: Suppose we use a significance level of 0.05 for the t-test. We can calculate the p-value using a t-
distribution table or software. Using the values from the previous examples, we get a p-value of 0.056.
Since the p-value is greater than the significance level, we fail to reject the null hypothesis that the
population mean test score is 85.