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

The null hypothesis represents the default claim.

\[ H_0 : \mu = \mu_0 \]

The alternative hypothesis represents the competing claim.

\[ H_a : \mu \neq \mu_0 \]

A test statistic measures how far the sample result is from the null hypothesis.

\[ z = \frac{\bar{x}-\mu_0}{\sigma/\sqrt{n}} \]

where:

\(\bar{x}\) = sample mean

\(\mu_0\) = hypothesized mean

\(\sigma\) = population standard deviation

\(n\) = sample size

The normal distribution is useful in hypothesis testing because many test statistics follow an approximately normal shape.

Suppose we collected the following exam scores from a sample of students.

##   Student Score
## 1       A    72
## 2       B    85
## 3       C    91
## 4       D    78
## 5       E    88

The following code calculates the average score from our sample data.

sample_data <- data.frame(
  Student = c("A", "B", "C", "D", "E"),
  Score = c(72, 85, 91, 78, 88)
)

mean(sample_data$Score)

## [1] 82.8

• Hypothesis testing helps determine whether there is enough evidence to support a claim.
• The null and alternative hypotheses provide the framework for statistical decision making.
• Test statistics and probability distributions help evaluate sample results.
• R and Plotly can be used to analyze data and create both static and interactive visualizations.
• These tools allow us to communicate statistical findings clearly and effectively.