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Understanding hypothesis testing is crucial for anyone delving into the realm of statistics. At its core, hypothesis testing is a method used to decide whether there is enough evidence to reject a null hypothesis, represented as \( H_0 \). The null hypothesis typically suggests that there is no effect or no difference, and it serves as a default or starting assumption to be challenged.The process of hypothesis testing involves proposing an alternative hypothesis, \( H_a \), which is a statement we are seeking evidence for, often suggesting that there is an effect or a difference. In the given exercise, we see an example where \( H_a: \mu eq 245 \) signifies that the mean is not equal to 245.To conduct a hypothesis test, one calculates a test statistic, which is then compared against a critical value or converted into a p-value for making a decision. If the p-value is less than a predetermined significance level, often \( \alpha = 0.05 \), we reject the null hypothesis. Conversely, if the p-value is larger, we fail to reject the null hypothesis. This procedure offers a systematic way to make decisions that account for uncertainty and variability in data.

A z-score provides a way of describing how far and in what direction, a data point is from the mean, measured in terms of standard deviations. It is a standardized score that helps to compare results from different tests or analyze individual scores within the same test.The formula to compute the z-score of a data point is:\[ z = \frac{(X - \mu)}{\sigma} \]where \( X \) represents the data point, \( \mu \) is the mean, and \( \sigma \) is the standard deviation. In our problem, we're given a z-score of 1.1, which means that the data point in question lies 1.1 standard deviations above the mean of the standard normal distribution.

The standard normal distribution, a special case of the normal distribution, is central to statistics. It has a mean of 0 and a standard deviation of 1. When a dataset follows a normal distribution, converting the data to z-scores translates the data into the standard normal distribution.Useful properties of the standard normal distribution include its symmetry and the fact that percentages of data falling within a certain number of standard deviations from the mean are fixed. For instance, about 68% of the data falls within 1 standard deviation from the mean, and approximately 95% falls within 2 standard deviations.In hypothesis testing, we often use the standard normal distribution to find p-values, as it allows us to determine the probability of observing a test statistic as or more extreme than the one calculated.

A two-tailed test is a statistical test where the area of interest is in both tails of the distribution. This test is used when the alternative hypothesis specifies a value that is different from the null hypothesis mean, meaning it could be either greater than or less than that mean but not equal to it.In such tests, we are looking for evidence against the null hypothesis in either direction. Accordingly, we split our significance level between both tails of the distribution. If \( z_* \) is the calculated test statistic from our data, to find the p-value, we determine the probability of seeing a z-score as extreme as \( z_* \) or more so in both directions, which involves doubling the p-value obtained from one tail, as seen in the exercise.Two-tailed tests are the appropriate choice when we don't have a specific direction of interest beforehand, as they provide a more conservative estimate compared to a one-tailed test by accounting for extremities in the data in either direction.