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The population mean, often represented as \( \mu \), is a central concept in statistics. It is the average of all values in a given population. In hypothesis testing, we're typically interested in making inferences about this mean based on sample data. This mean tells us about the overall central tendency of a whole group of data and is a fundamental aspect of probability and statistical analysis.

In the context of hypothesis testing, we use the population mean to form hypotheses. For instance, when testing whether the population mean equals 5, our null hypothesis is stated as \( H_0: \mu = 5 \). Any deviation from this mean in our hypothesis testing would indicate whether or not we should accept or reject this hypothesis. The known variance \( \sigma^2 \) of the population also plays a crucial role in assessing how much the values are spread out around the mean. Together with the mean, the variance helps in determining the likelihood of the observed sample belonging to the population with the stated mean.

The significance level, symbolized by \( \alpha \), is a threshold in hypothesis testing. It's the probability of rejecting the null hypothesis when it is actually true, also known as the Type I error probability. This value is pre-determined by the researcher and often set at common levels such as 0.01, 0.05, or 0.10.

A lower significance level means you require stronger evidence against the null hypothesis to reject it. For instance, at \( \alpha = 0.01 \), there's only a 1% risk of claiming a statistically significant result when there isn't one, requiring more conservative evidence compared to a 10% risk at \( \alpha = 0.10 \). In our context, we are considering significance levels of 0.01, 0.05, and 0.10, each influencing the determination of our critical value.

A critical value helps determine the borderline between the acceptance region and the rejection region for the null hypothesis. In hypothesis testing, it is derived based on the chosen significance level. The critical value depends on the nature of the test. For a one-tailed test like the left-tailed tests we're considering, the critical value is identified from a standard normal distribution (Z-distribution).

To find the critical value for significance levels 0.01, 0.05, and 0.10, we use a Z-table or statistical software. For the scenarios described:

• At \( \alpha = 0.01 \), the critical value is approximately \( Z_0 = -2.33 \).
• At \( \alpha = 0.05 \), the critical value is approximately \( Z_0 = -1.645 \).
• At \( \alpha = 0.10 \), the critical value is approximately \( Z_0 = -1.28 \).

These values tell us how extreme the test statistic must be in order to reject the null hypothesis at each significance level.

The Z-statistic is a vital component of hypothesis testing when dealing with known population variance. It quantifies how many standard deviations an element is from the mean. In a hypothesis test, we compute the Z-statistic using the formula: \[Z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}\]Where:

• \( \bar{x} \) is the sample mean.
• \( \mu \) is the population mean stated in the null hypothesis.
• \( \sigma \) is the population standard deviation.
• \( n \) is the sample size.

In testing whether a population mean is less than a specified value, we compare the computed Z-statistic against the critical value derived for our chosen significance level. A Z-statistic that falls below the critical value indicates sufficient evidence to reject the null hypothesis. This process ensures that we base our decision not on random chance but on the statistical analysis of our sample data.