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The standard normal distribution is a critical concept in statistics. It is a probability distribution that is symmetrical and bell-shaped. The center of this distribution is situated at the mean, which is zero, and the standard deviation is one. Understanding this distribution is crucial for many statistical tests, including calculating p-values.

Some important characteristics of the standard normal distribution include:

• It is defined by its mean (\( \mu = 0 \)
• It is defined by its standard deviation (\( \sigma = 1 \)
• It is a continuous distribution, so every value on the horizontal axis (z-score) has a probability associated with it.

When we discuss the standard normal distribution, one important tool is the z-table or statistical software that helps find the p-values. These tools show the probability of a standard normal variable being less than a given value of z.

For instance, if you have a z-score of 0.84, you can use the standard normal table to locate the probability associated with it for one side of the distribution. Using a standard normal distribution in hypothesis testing helps to determine how exceptional an observed outcome is, compared to what is expected under the null hypothesis.

An upper tail test, also known as a right-tailed test, is a type of hypothesis test. It is used to determine if a particular statistic is significantly higher than what is expected under the null hypothesis. For example, if we're testing whether a proportion is greater than a hypothesized value, this test will help us understand that.

• The null hypothesis (\( H_0 \)) typically states that the parameter is less than or equal to a value.
• The alternative hypothesis (\( H_1 \)) states that the parameter is greater than that value.
• The p-value is calculated using the probability of obtaining a result as extreme as or more extreme than the observed statistic, assuming the null hypothesis is true.

Consider a scenario where we have a z-score of 0.84. For an upper tail test, we look for the area to the right of this value. This area represents the probability that the observed data (or something more extreme) would occur if the null hypothesis were true. Thus, for the z-score, the p-value is \( P(Z > 0.84) \), indicating the right side of the distribution.

A two-tailed test is employed when we want to detect any significant deviation from the null hypothesis, without specifying the direction. It's commonly used when deviations could happen in two directions, either greater or lesser than the hypothesized parameter.

Here's how a two-tailed test works:

• The null hypothesis (\( H_0 \)) states there is no effect or difference.
• The alternative hypothesis (\( H_1 \)) implies there is a difference, but it doesn't specify if the test statistic is greater or less than the hypothesized value.
• The p-value, in this context, is calculated by adding the probabilities from both tails of the distribution that are as extreme as the observed z-score.

Take, for instance, a z-score of 2.25. In a two-tailed test, the probability of finding such an extreme value is calculated on both ends of the distribution. We add the probabilities \( P(Z < -2.25) \) and \( P(Z > 2.25) \). Alternatively, since the normal distribution is symmetric, we can multiply one tail’s probability by two, thus \( 2P(Z > 2.25) \), which gives the comprehensive p-value. This encompasses the extremity on either side.