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The z-score is a statistical measure that tells you how many standard deviations a point is from the mean of a data set. It helps in determining how far and in which direction a sample deviates from the population mean. To compute the z-score, you subtract the population mean from the sample mean and then divide by the standard error (which is the standard deviation divided by the square root of the sample size). In the exercise, two different sample means are evaluated. For the first sample, the calculation

• Subtracts 45 from 41.8 to get -3.2.
• Divides -3.2 by 1.2 (where 1.2 is obtained by dividing the population standard deviation of 6 by the square root of 25, which is 5).
• This gives a z-score of -2.67.

For the second sample with a mean of 43.8, the z-score is calculated as -1. This z-score helps in comparing the sample mean to the theoretical mean and deciding on the null hypothesis.

The null hypothesis, denoted as \(H_0\), is a statement that posits no effect or no difference and serves as the starting assumption for hypothesis testing. In scientific studies, the null hypothesis is typically something the researcher wants to test against to prove a new theory or idea. In the given exercise, the null hypothesis is formulated as \(H_{0}: \mu=45\), suggesting that the true population mean is 45.

• This baseline assumption allows us to use methods like z-scores to test whether the evidence from a sample is strong enough to reject this hypothesis.
• If the z-score falls into a certain critical region determined by the confidence level (in this case, a significance level \(\alpha = 0.025\)), it may lead us to reject or fail to reject this null hypothesis.

Understanding and establishing a null hypothesis correctly is crucial for conducting valid hypothesis testing.

The alternative hypothesis, denoted as \(H_1\), suggests an opposite statement to the null hypothesis that the researcher wants to test. It signifies the presence of an effect or a difference and is what researchers are typically trying to prove. In the exercise, the alternative hypothesis is \(H_{1}: \mu<45\), indicating that the researcher suspects the true population mean to be less than 45.

• This hypothesis is formulated based on the assumption that the sample evidence will support this direction.
• It forms the basis for conducting the hypothesis test alongside \(H_0\).

A hypothesis test ultimately determines whether there is enough statistical evidence in favor of this alternative hypothesis. If data suggest that the sample mean significantly deviates from the value proposed in the null hypothesis, then it supports the alternative hypothesis.

The critical value is a point beyond which we would reject the null hypothesis in favor of the alternative hypothesis. It is determined by the significance level \(\alpha\) and the type of test being conducted (one-tail or two-tail). In this exercise, a one-tailed test is used with an \(\alpha\) of 0.025, leading to a critical value of about -1.96.

• If the computed z-score is less than this critical value, it indicates the result is statistically significant, and we reject \(H_0\).
• For the first sample mean resulting in a z-score of -2.67, the result lies outside in the rejection region, which leads us to reject the null hypothesis.
• For the second sample mean with a z-score of -1.0, it does not surpass the critical boundary, hence we fail to reject the null.

Choosing and understanding the role of a critical value is crucial as it helps determine the statistical significance of a test result.