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The standard normal distribution is a bell-curved distribution that is symmetric about the mean, which is zero for this specific distribution, and has a standard deviation of one. It is a special case of normal distributions used for standardized testing in statistics. Any normal distribution can be converted into the standard normal distribution by using the transformation \( Z = \frac{X - \mu}{\sigma} \), where \( Z \) is the standard normal variable, \( X \) is the variable being standardized, \( \mu \) is the mean, and \( \sigma \) is the standard deviation of the original distribution.

The area under the curve of the standard normal distribution represents probability. Since the total area under the curve is 1, which corresponds to 100% probability, the area to the left of a given z-score gives the probability that a value will be less than that z-score. This is foundational for calculating p-values and making inferences in hypothesis testing.

A left-tail test in statistics is a hypothesis test where the critical region, where we would reject the null hypothesis, is in the left tail of the distribution. This type of test is used when we want to determine if a population parameter is less than a given value.

For example, in the solved exercise, the test statistic calculated, \( z = -1.08 \), is used to find the p-value by looking up the cumulative probability of \( z \) being less than or equal to -1.08 in the standard normal distribution table. The p-value is the area to the left of \( z \) which is 0.1401, indicating the probability of obtaining a test statistic as extreme as -1.08 or more extreme, given that the null hypothesis is true.

A right-tail test is the opposite of a left-tail test. It is when we're interested in determining if a population parameter is greater than a certain value. The critical region in a right-tail test is on the right side of the distribution curve.

In the provided exercise, the p-value calculation for the right-tail test uses a test statistic of \( z = 4.12 \). We calculate the p-value by subtracting the cumulative probability of \( z \) being less than or equal to 4.12 from 1 (since we want the area to the right of \( z \)). Since \( P(Z \leq 4.12) \) is very close to 1, it makes the p-value almost 0, indicating a very low probability of observing such an extreme statistic under the null hypothesis.

In a two-tailed test, the critical regions are on both tails of the distribution curve, which is used when we want to test if a population parameter is significantly different from a given value, regardless of the direction. The p-value in this scenario is the probability of observing a test statistic as extreme as the one calculated or more extreme, in either direction.

For the test statistic \( z = -1.58 \), we calculate the p-value by first finding the area to the right of \( 1.58 \) which is then multiplied by 2 to account for both tails of the distribution, yielding a p-value of approximately 0.114. This reflects the combined probability of observing a test statistic of less than -1.58 or greater than 1.58 under the null hypothesis.

The test statistic is a standardized value calculated from sample data during a hypothesis test. It is used to determine the likelihood of a given hypothesis, by comparing the test statistic to a critical value from a distribution. In the context of the normal distribution, the test statistic is often a z-score which helps to determine the distance of sample data from the hypothesized population parameter in terms of standard deviation units.

Each of the p-value examples given in the solution steps involves calculating a z-score as the test statistic, which is then used to find the corresponding probabilities in the standard normal distribution to determine p-values.

The cumulative distribution function, or CDF, for a random variable gives the probability that the variable will take a value less than or equal to a specific value. For the standard normal distribution, the CDF represents the area under the curve to the left of a z-value.

In the case of p-value calculations, using the cumulative distribution function of the standard normal distribution allows us to find the exact probability associated with our test statistic. It cumulatively adds up all probabilities from the far left of the distribution up to the z-score we are evaluating, providing a full picture of the likelihood of observing a test statistic at least as extreme as the one calculated.