Hypothesis Testing is a statistical method to decide whether the data sufficiently supports a particular hypothesis.

\[ H_0: \text{Null Hypothesis} \] \[ H_A: \text{Alternative Hypothesis} \]

• Null Hypothesis (H₀): The assumption that there is no difference between groups or variables being studied.
• Alternative Hypothesis (H₁): The statement that researchers are trying to prove (often the same as the research hypothesis).
• Significance Level (α): The probability that the event could have occured by chance (commonly 0.05).
• p-value: The number describing the likelihood of obtaining the observed data under the hull hypothesis.

We use the mtcars dataset to test if the mean miles per gallon (MPG) is significantly different from 20.

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##                    mpg cyl disp  hp drat    wt  qsec vs am gear carb
## Mazda RX4         21.0   6  160 110 3.90 2.620 16.46  0  1    4    4
## Mazda RX4 Wag     21.0   6  160 110 3.90 2.875 17.02  0  1    4    4
## Datsun 710        22.8   4  108  93 3.85 2.320 18.61  1  1    4    1
## Hornet 4 Drive    21.4   6  258 110 3.08 3.215 19.44  1  0    3    1
## Hornet Sportabout 18.7   8  360 175 3.15 3.440 17.02  0  0    3    2
## Valiant           18.1   6  225 105 2.76 3.460 20.22  1  0    3    1

Performing a one-sample t-test:

## 
##  One Sample t-test
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## data:  mtcars$mpg
## t = 0.08506, df = 31, p-value = 0.9328
## alternative hypothesis: true mean is not equal to 20
## 95 percent confidence interval:
##  17.91768 22.26357
## sample estimates:
## mean of x 
##  20.09062

The test statistic follows:

\[ t = \frac{\bar{x} - \mu}{s/\sqrt{n}} \]

where: - \(\bar{x}\) is the sample mean - \(\mu\) is the hypothesized mean - \(s\) is the sample standard deviation - \(n\) is the sample size

Hypothesis testing is a useful tool that aids our decision-making. Making a data-informed decision is the best choice.