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The standard normal distribution is a fundamental concept in statistics. It is a special kind of normal distribution with a mean of 0 and a standard deviation of 1. This distribution is very useful because it allows comparisons across different tests and measurements. When we say that a test statistic, like a Z-score, follows a standard normal distribution, we mean that it can be compared against this curve. The Z-score itself measures how many standard deviations away a particular value is from the mean. In hypothesis testing, the standard normal distribution is used to determine the probability of observing a test statistic as extreme, or more extreme than, the observed value if the null hypothesis is true. This probability is crucial in deciding whether to reject or fail to reject the null hypothesis.

The significance level, denoted as \( \alpha \), plays a key role in hypothesis testing. It represents the probability of rejecting the null hypothesis when it is actually true, which is known as a Type I error. Commonly chosen significance levels are 0.05, 0.01, or 0.10, depending on the study's requirements. When planning an experiment, researchers decide on the significance level beforehand to ensure that the test's results are interpreted correctly. In the context of our exercise:

• In case (a), the significance level is calculated by finding the probability that the test statistic is greater than or equal to 1.88, which was 0.0301.
• In case (b), it was 0.0030, highlighting a much stricter test criterion.
• In case (c), we dealt with a two-tailed test with a significance level of 0.0040.

Understanding and choosing the significance level is crucial as it informs how confident you can be in your test results.

The Z-table, also known as the standard normal table, is a tool that helps you find the probability that a standard normal random variable is less than or equal to a given value. In simpler terms, it tells you how likely it is for a score to be below a certain point on the standard normal distribution curve. Using the Z-table involves looking up a Z-score on the table, which then gives you the cumulative probability up to that Z-score. This is particularly helpful when determining significance levels or critical values in hypothesis testing. For example, in step 2 of our solution:

• We used the Z-table to find that \( P(Z < 1.88) = 0.9699 \).
• This probability helped us calculate the significance level \( \alpha = 1 - 0.9699 = 0.0301 \).

By understanding how to use a Z-table, you can effectively find the necessary probabilities to conduct hypothesis tests.

The concept of a rejection region is integral to hypothesis testing. It defines the range of values for which we reject the null hypothesis. If the test statistic falls into this region, the observed result is considered statistically significant. In the context of our problem:

• For a right-tailed test like in case (a), the rejection region was defined as \( z \geq 1.88 \).
• In a left-tailed test like case (b), the rejection region was \( z \leq -2.75 \).
• A two-tailed test, like in case (c), involves rejection regions at both ends: \( z \geq 2.88 \) and \( z \leq -2.88 \).

The rejection region is determined based on the significance level. It helps decide whether the statistical evidence is strong enough to reject the null hypothesis.