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The one-sample z test is a statistical method used to determine if a sample mean is significantly different from a known or hypothesized population mean. This type of test is ideal when you have a normally distributed population and know the population standard deviation, denoted as \( \sigma \). For example, in our exercise, we want to test whether the population mean is 4.5 against the alternative hypothesis that it is greater than 4.5.

This involves several steps:

• Setting up the null and alternative hypotheses.
• Calculating the z-score: \({ z = \frac{( \overline{x} - \mu )}{ ( \frac{\sigma}{\sqrt{n}} ) } }\)

In our case, the sample mean \( \overline{x} = 4.9 \), the population mean \( \mu = 4.5 \), the population standard deviation \( \sigma = 1.2 \), and the sample size \( n = 13 \). By substituting these values, we calculate the z-score.

The p-value helps us understand the probability of obtaining a sample mean as extreme or more extreme than the observed one, assuming the null hypothesis is true. A small p-value (typically less than 0.05) indicates strong evidence against the null hypothesis.

To find the p-value:

• Compute the z-score as \({ z = \frac{(4.9 - 4.5)}{(\frac{1.2}{\sqrt{13})}} = 1.148} \)
• Use a standard normal distribution (Z-table) to find the probability corresponding to z = 1.148.

This probability represents the p-value, which tells us how likely it is to observe a sample mean of 4.9 if the true population mean is 4.5.

The level of significance, denoted as \( \alpha \), is a threshold set by the researcher to decide whether to reject the null hypothesis. Common values for \( \alpha \) are 0.05, 0.01, and 0.10, depending on the context and desired stringency of the test.

In our example, we use \( \alpha = 0.1 \). This means that the researcher is willing to accept a 10% chance of incorrectly rejecting the null hypothesis. To make a decision:

• Compare the p-value to \( \alpha \).
• If the p-value < \( \alpha \), reject the null hypothesis.
• If the p-value \( \geq \alpha \), do not reject the null hypothesis.

The null hypothesis (\( H_0 \)) is a statement that there is no effect or no difference, and it serves as a starting point for statistical testing. In our exercise, the null hypothesis states that the population mean \( \mu = 4.5 \). We use the null hypothesis to calculate the z-score and p-value.

The logic of hypothesis testing works as follows:

• Assume \( H_0 \) is true until we have strong evidence against it.
• Compute the test statistic and p-value based on \( H_0 \).
• Make a decision to either reject or not reject \( H_0 \), using the level of significance as a guide.

The alternative hypothesis (\( H_1 \) or \( H_a \)) is what you want to prove. It is a statement indicating that there is a statistically significant effect or difference. In our example, we state the alternative hypothesis as \( H_1: \mu > 4.5 \).

This means we are testing if the true population mean is greater than 4.5.

• If our p-value is smaller than our chosen significance level, we have enough evidence to support \( H_1 \).
• The decision rule can be summarized: reject \( H_0 \) if p-value < \( \alpha \); otherwise, do not reject \( H_0 \).

By following these steps attentively, one can properly conduct hypothesis testing and interpret the results accurately.