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The one-sample t-test allows you to determine if a sample mean is significantly different from a known value, often a population mean.
It is useful when comparing the means of a single group against a theoretical value.
Here's how it works:

• Take a sample from your data set. For instance, consider the sample values: 18, 15, 12, 19, and 21.
• Calculate the sample mean and standard deviation.
• Then, compute the t-test statistic, which assesses how far the sample mean is from the population mean, scaled by the variability of the sample data.

In the exercise, we used the formula:\[t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}\]Where:- \( \bar{x} \) is the sample mean- \( \mu_0 \) is the hypothesized population mean (20 in this case)- \( s \) is the sample standard deviation- \( n \) is the sample sizeThis test assumes that the sample data is approximately normally distributed and is particularly useful when dealing with smaller samples.

The significance level, often denoted by \( \alpha \), is the threshold used to determine when to reject the null hypothesis.
A common choice is 0.05, but in our case, we use a stricter level of 0.01, indicating a high confidence requirement.
What does this mean?

• It's the probability of rejecting the null hypothesis when it is actually true (Type I error).
• Lower significance levels reflect higher standards for evidence against the null hypothesis.

In practice:- A significance level of 0.01 means there's only a 1% risk of mistakenly rejecting the null hypothesis.This standard helps ensure the results are not due to random chance, and the chosen significance level should reflect the consequences of error in decision-making.

The null hypothesis (\( H_0 \)) is a statement that there is no effect or no difference, and it is the default position in hypothesis testing.
In the context of the exercise, the null hypothesis is that the population mean \( \mu \) is greater than or equal to 20.
Why is it important?

• The null hypothesis serves as a baseline that helps us determine if the observed data is significantly different.
• It is what we assume to be true unless we have evidence to prove otherwise.
  
• Testing the null hypothesis involves conducting a t-test and comparing the resulting p-value to the chosen significance level.

Only when evidence against \( H_0 \) is strong enough, do we reject it in favor of an alternative hypothesis (\( H_1 \)). It helps provide a structured approach to statistical inference.

The p-value is a measure that helps you determine the significance of your results in hypothesis testing.
It tells us how likely it is to observe data at least as extreme as ours under the null hypothesis \( H_0 \).
Understanding the p-value:

• It ranges from 0 to 1. A smaller p-value means stronger evidence against \( H_0 \).
• It's compared to the significance level \( \alpha \) to decide whether to reject \( H_0 \).

In detail:- A p-value less than \( \alpha \) means you reject \( H_0 \), supporting the alternative hypothesis.- In our exercise, the estimated p-value was more than 0.01, which was not sufficient to reject \( H_0 \).The p-value gives a tangible measure of how extreme the sample results are, considering the null hypothesis is true. It's a crucial part of statistical hypothesis testing, providing insight into the determination of the research outcome.