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In hypothesis testing, we begin by setting up two hypotheses: the null hypothesis and the alternative hypothesis. These act as the foundation for statistical testing.
The **Null Hypothesis** (\( H_0 \)) is a statement that there is no effect or no difference, and it proposes that any difference observed is due to sampling or random chance. In our example, the null hypothesis is: "The car's true average fuel efficiency is at least 30 mpg," or mathematically, \( \mu \geq 30 \).
On the other hand, the **Alternative Hypothesis** (\( H_1 \)) is what you might want to prove. It indicates that the effect or difference you’re testing is real. For our automobile test, the alternative hypothesis is: "The car's true average fuel efficiency is less than 30 mpg," or \( \mu < 30 \).
Setting these hypotheses clearly helps establish a binary structure where you either reject the null hypothesis in favor of the alternative or you fail to reject the null hypothesis.

The t-test is a statistical test used to evaluate the difference between the sample mean and the population mean. In our scenario, we use it to determine whether the car's reported fuel efficiency of 30 mpg stands true with the sample data.

To perform a t-test:

• Calculate the **sample mean** (\( \bar{x} \)) and **sample standard deviation** (\( s \)). These statistics provide an overview of the data's central tendency and dispersion.
• Determine the **t statistic** using the formula: \( t = (\bar{x} - \mu) / (s / \sqrt{n}) \). Here, \( \bar{x} \) is the sample mean, \( \mu \) is the claimed population mean (30 mpg), \( s \) is the sample standard deviation, and \( n \) is the sample size.
• This t statistic helps to compare the sample and population mean and determine if the observed differences are statistically significant.

The t-test assumes that the sample is drawn from a normally distributed population, which fits the conditions here.

The p-value is a probability that measures evidence against the null hypothesis. It tells us the likelihood of observing our sample data, assuming the null hypothesis is true.

To interpret the p-value:

• After calculating the t statistic, refer to a **t-distribution table** to find the p-value.
• Compare this p-value with a pre-defined significance level, typically 0.05.
• If the **p-value** is less than the significance level, it indicates strong evidence against the null hypothesis, leading us to reject it.
• In contrast, a **p-value** greater than the significance level suggests insufficient evidence to reject the null hypothesis.

For our test, a low p-value would suggest rejecting the claim that the car averages at least 30 mpg, while a higher p-value would mean that our sample data does not contradict this claim.