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The null hypothesis, denoted as \( H_{0} \), is a statement that there is no effect or no difference. It serves as the starting assumption for hypothesis testing. We assume the null hypothesis is true until we have enough evidence to support the alternative hypothesis.
In the given exercise, the null hypothesis is \( H_{0}: \sigma = 4.2 \). This states that the population standard deviation \( \sigma \) is equal to 4.2. The null hypothesis typically represents a status quo or a benchmark against which we compare our findings. If our test provides sufficient evidence, we may reject \( H_{0} \) in favor of the alternative hypothesis.

The alternative hypothesis, denoted as \( H_{1} \), is a statement that contradicts the null hypothesis. It represents what we suspect might be true instead. This hypothesis basically indicates that there is an effect or a difference.
For our exercise, the alternative hypothesis is \( H_{1}: \sigma eq 4.2 \). This suggests that the population standard deviation is not equal to 4.2, so it could be either less than or greater than 4.2. The alternative hypothesis is what we are trying to provide evidence for by rejecting the null hypothesis in favor of \( H_{1} \).
In hypothesis testing, we always frame both null and alternative hypotheses before conducting any test.

A two-tailed test checks for any significant difference in either direction from the stated null hypothesis. Unlike a one-tailed test that looks for an effect in one specific direction (either greater than or less than), a two-tailed test examines both.
In the context of our exercise, the two-tailed test is relevant because our alternative hypothesis \( H_{1}: \sigma eq 4.2 \) contends that the population standard deviation is not equal to 4.2 – meaning it could be higher or lower. Therefore, we need to consider possible deviations in both directions.
To interpret the results of a two-tailed test, if our test statistic falls into the critical region on either tail of the distribution, we reject the null hypothesis.

The population standard deviation, denoted by \( \sigma \), measures the dispersion or spread of a set of data points in a population. It tells us how much the individual data points typically deviate from the mean of the population.
In the exercise, the hypotheses \( H_{0}: \sigma = 4.2 \) and \( H_{1}: \sigma eq 4.2 \) focus on testing whether the actual population standard deviation is 4.2 or any value but 4.2.
Understanding the population standard deviation helps in assessing the variability within the data and is crucial for hypothesis testing, as it impacts the calculation of the test statistic and confidence intervals. Standard deviation is a key parameter frequently tested in statistics to understand the spread and reliability of our data.