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STAT101 lecture notes
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Hypothesis testing is an essential tool in statistical analysis, enabling us to make decisionsabout the population based on the sample data. The null hypothesis, which is the starting point
of the hypothesis testing, assumes that there is no significant difference or association between
the variables of interest. On the other hand, the alternative hypothesis challenges the null
hypothesis and states that there is a significant difference or association between the variables.
To test the null hypothesis, we need to calculate the p-value, which is the likelihood of observing
data at least as extreme as what was observed, assuming that the null hypothesis is true. In this
lecture, we will discuss the p-value in detail, including its definition, interpretation, and
calculation.
Definition of P-Value:
The p-value is a probability value that measures the evidence against the null hypothesis. It is
the probability of observing a test statistic as extreme as the one calculated from the sample
data, assuming that the null hypothesis is true. The test statistic is a summary statistic
calculated from the sample data that measures the difference between the observed data and
the null hypothesis.
The p-value ranges from 0 to 1, with values closer to 0 indicating strong evidence against the
null hypothesis, and values closer to 1 indicating weak evidence against the null hypothesis. In
hypothesis testing, we set a significance level, denoted by alpha, which is the maximum
probability of making a Type I error, rejecting the null hypothesis when it is actually true.
Common values of alpha are 0.05 or 0.01, indicating a 5% or 1% chance of making a Type I
error, respectively.
Interpretation of P-Value:
The interpretation of the p-value depends on the significance level and the research question. If
the p-value is less than or equal to the significance level, we reject the null hypothesis and
conclude that there is sufficient evidence to support the alternative hypothesis. In other words,
we reject the idea that the observed difference or association is due to chance and instead
conclude that there is a significant difference or association between the variables of interest.
On the other hand, if the p-value is greater than the significance level, we fail to reject the null
hypothesis and conclude that there is insufficient evidence to support the alternative hypothesis.
In other words, we do not have enough evidence to claim that there is a significant difference or
association between the variables of interest.
It is essential to note that failing to reject the null hypothesis does not necessarily mean that the
null hypothesis is true. It simply means that we do not have enough evidence to reject it.
Furthermore, a p-value greater than the significance level does not necessarily mean that the
null hypothesis is true, as it is possible to obtain a p-value greater than the significance level
due to chance variability in the sample data.
Calculation of P-Value:
The calculation of the p-value depends on the type of test statistic and the null hypothesis.
There are three common methods for calculating the p-value, namely, the z-test, t-test, and
chi-square test.
Z-Test:
The z-test is used when the population standard deviation is known. The test statistic is
calculated as:
z = (x - mu) / (sigma / sqrt(n))
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