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In hypothesis testing, the critical value is a threshold that determines if we reject or fail to reject the null hypothesis. It's directly related to the significance level, \(\alpha\), which represents the probability of making a Type I error or rejecting a true null hypothesis.
To find the critical value for a test with a given significance level, you typically consult a standard normal distribution table (Z-table). This table provides the Z-score that corresponds to your chosen significance level. For a left-tailed test at \(\alpha=0.01\), the critical value is negative since it represents the lower tail of the distribution.
In the two-tailed test context, such as in scenario b) where \(H_1: p eq 0.63\), you split the significance level between two tails, meaning you calculate critical values for \(\pm \alpha/2\). You'll reject the null hypothesis if the observed test statistic falls beyond these critical values, indicating that the sample data is significantly different from the presumed population parameter.

The z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. In hypothesis testing, it transforms the sample proportion into a standard score considering both the sample size and the variability under the null hypothesis.
To calculate the z-score, we use the formula: \[Z = \frac{(p - P_{0})*\sqrt{n}}{\sqrt{P_{0}*(1-P_{0})}} \]where:

• \(p\) is the observed sample proportion,
• \(P_{0}\) is the hypothesized population proportion,
• \(n\) is the sample size.

The z-score indicates how many standard deviations the observed sample proportion is from the hypothesized proportion under the null hypothesis. In a practical sense:

• If the z-score falls within the critical value(s), we do not reject the null hypothesis.
• If it falls outside, we reject the null hypothesis, suggesting the sample provides sufficient evidence against the null hypothesis.

In statistics, the sample proportion is a crucial concept when estimating the proportion of a characteristic in a population from a sample. It is simply the number of favorable cases divided by the total sample size.
Using the exercise as an example, the sample proportion is \(0.60\) since 120 observations of the 200 met the criteria being evaluated. The sample proportion offers a point estimate of the population proportion, giving you a good idea of what the corresponding population proportion might be.
However, since the sample proportion is based on a subset of the population, it's important to recognize the role of variability and the potential for sampling error. This is accounted for when calculating the standard error of the proportion and applying the z-score formula in hypothesis testing.

The alternative hypothesis, denoted as \(H_1\), is a statement that suggests a potential outcome that contradicts the null hypothesis. It reflects what we might suspect to be true or are trying to prove.
In the case of statistical tests, the alternative hypothesis is key because it defines the region of rejection. For example:

• For part (a) of the exercise, \(H_1: p < .63\), we have a one-tailed test that looks for the sample proportion being significantly less than \(0.63\). The critical region is thus in the left tail of the distribution.
• For part (b), where \(H_1: p eq 0.63\), it is a two-tailed test, meaning we are interested in deviations in both directions from \(0.63\).

Selecting an alternative hypothesis is not arbitrary; it stems from theoretical prediction, prior research, or specific inquiry goals. Effectively, it guides the interpretation of statistical test results and helps decide whether to reject the null hypothesis.