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Normal distribution is a fundamental concept in statistics, often described as a bell-shaped curve. It is a continuous probability distribution characterized by:

• a mean (\( \mu \)) which determines the center of the distribution
• a variance (\( \sigma^2 \)) which influences the width or spread of the curve

In this exercise, we have a sample taken from a population that follows a normal distribution. The normal distribution is symmetric about its mean, and it’s used frequently due to its excellent properties related to the Central Limit Theorem.
When dealing with normal distribution, it's important to understand that it is fully described by its mean and variance. Hence, even if only the mean or variance is unknown, the distribution itself remains fully predictable given the other parameter is known.
In hypothesis testing of this exercise, we exploit the properties of normal distribution to assess whether the sample came from a population with a specific mean.

The z-test is a statistical method employed when dealing with data assumed to follow a normal distribution with a known variance. It helps to determine whether there's a significant difference between the sample mean and a known population mean.
When executing a z-test:

• We calculate a statistic called the z-value or z-score using the formula \[ Z = \frac{\bar{Y} - \mu_0}{\sigma/\sqrt{n}} \]where \( \bar{Y} \)is the sample mean, \( \mu_0 \) is the population mean under the null hypothesis, \( \sigma \) is the standard deviation, and \( n \) is the sample size.
• The calculated z-score tells us how many standard deviations the sample mean is from the population mean.
• We compare this z-score to a critical value from the z-table to determine whether we reject the null hypothesis.

In this context, we use the z-test to check if our sample mean significantly deviates from the hypothesized population mean of 7.

The power function in hypothesis testing is a crucial concept that measures the test's ability to correctly reject a false null hypothesis. It is the probability that the test will lead to a rejection of \( H_0 \) when an alternative hypothesis \( H_a \) is true.
Power is especially important because:

• It quantifies the effectiveness of a test.
• A higher power means a higher probability of detecting a true effect if one exists.
• The power is affected by factors such as sample size, effect size, significance level, and variance.

In the exercise, the power function helps illustrate the test's ability to reject \( H_0 \) for several alternative values of \( \mu \). By plotting the power against alternative means, we see how it increases as \( \mu \) moves away from \( H_0 \), demonstrating the test's increasing sensitivity.

The significance level, denoted as \( \alpha \), is a threshold used in hypothesis testing to determine whether a result is statistically significant. It represents the probability of rejecting the null hypothesis when it is actually true, which is also known as a Type I error.

• Common values for \( \alpha \) are 0.05, 0.01, and 0.10. In this exercise, it's set to 0.05.
• It defines the critical region: test results falling in this region lead to rejection of \( H_0 \).
• A smaller \( \alpha \) indicates stricter criteria for significance but increases the risk of Type II error, which is failing to reject a false null hypothesis.

Choosing the right significance level is a balance: lower \( \alpha \) means more confidence in the results, while higher \( \alpha \) means easier rejection of \( H_0 \). For this problem, the critical z-value corresponding to the 0.05 significance level is approximately 1.645, establishing the boundary for making a statistical decision on \( H_0 \).