Hypothesis testing is a statistical method used to determine whether there is enough evidence in a sample to support a claim about a population.

It is commonly used in science, engineering, business, and medicine to make data-driven decisions.

Hypothesis testing helps answer questions such as:

• Is a new teaching method more effective?
• Is the average product lifetime different from what is claimed?
• Did a treatment actually improve results?

It allows researchers to make conclusions using statistical evidence.

In hypothesis testing, we begin with two competing statements.

Null hypothesis:

\[ H_0 : \mu = 75 \]

Alternative hypothesis:

\[ H_a : \mu \neq 75 \]

Where:

• \(H_0\) = null hypothesis
  
• \(H_a\) = alternative hypothesis
  
• \(\mu\) = population mean

A commonly used test statistic is the z-score.

\[ z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}} \]

Where:

• \(\bar{x}\) = sample mean
  
• \(\mu\) = population mean
  
• \(\sigma\) = population standard deviation
  
• \(n\) = sample size

This statistic measures how far the sample mean is from the hypothesized population mean.

Suppose we want to test whether the average exam score is different from 75.

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   54.91   73.06   78.62   78.90   84.92   99.87

This dataset simulates 100 exam scores.

This histogram shows the distribution of simulated exam scores.

The boxplot shows the median, quartiles, and possible outliers in the data.

.

mean(scores)

## [1] 78.90406

sd(scores)

## [1] 9.128159

t.test(scores, mu = 75)

## 
##  One Sample t-test
## 
## data:  scores
## t = 4.2769, df = 99, p-value = 4.374e-05
## alternative hypothesis: true mean is not equal to 75
## 95 percent confidence interval:
##  77.09283 80.71528
## sample estimates:
## mean of x 
##  78.90406

This slide includes R code that calculates statistics and performs a hypothesis test.

Hypothesis testing is a powerful statistical tool used to evaluate claims about populations.

In this presentation we:

• defined hypothesis testing
  
• used mathematical formulas
  
• visualized data with ggplot
  
• created an interactive plotly visualization
  
• included statistical code in R

These techniques help analysts make informed, data-driven decisions.