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In hypothesis testing, we start by establishing two competing statements: the null hypothesis and the alternative hypothesis. The null hypothesis (H_0) is a statement of no effect, no difference, or a scientific hypothesis that a researcher attempts to disprove. For instance, if we have two income groups and we are investigating whether their approval for a product is the same, the null hypothesis would state that there is no significant difference between the approval percentages.The alternative hypothesis (H_1 or H_a), on the other hand, represents a statement contrary to the null hypothesis. It's what you might choose to believe if you reject the null hypothesis. In the same scenario, this could be the supposition that one income group has a higher approval percentage for a product than the other. It is essential to define these hypotheses precisely because they establish the basis of the testing procedure.

The test statistic is a standardized value that is calculated from sample data during a hypothesis test. It's used to determine the likelihood that the null hypothesis is true. In many cases, the test statistic follows a certain probability distribution under the assumption that the null hypothesis is correct. For comparing two proportions, as in the exercise, we often use the Z-test to compute a Z-score, which represents how many standard deviations a data point is from the mean.A/z-test statistic is usually calculated by taking the difference between the sample statistic and the null hypothesis, then dividing it by the standard error. The process is different depending on the type of data and the test being performed, but the aim is to quantify the weight of evidence against the null hypothesis.

A Z-score is a numerical measurement used in statistics to describe a value's relationship to the mean of a group of values, measured in terms of standard deviations from the mean. If a Z-score is 0, it indicates that the data point's score is identical to the mean score. A Z-score can be positive or negative, indicating the number of standard deviations it is above or below the mean, respectively. In hypothesis testing, the calculated Z-score is used as the test statistic for Z-tests. It allows us to determine how extreme or rare our data is under the null hypothesis and aids in deciding whether or not to reject the null hypothesis. High absolute values of Z-scores (both positive and negative) are associated with low probabilities of occurrence under the null hypothesis, which can be pivotal in the decision-making process of the test.

The level of significance (\(alpha\)), often represented as a percentage, is a critical part of hypothesis testing. It defines the threshold for how extreme the test statistic must be for us to reject the null hypothesis. It's essentially the probability of making the wrong decision when the null hypothesis is actually true (Type I error).In practice, common choices for \(alpha\) include 0.05 (5%), 0.01 (1%), and 0.10 (10%). The level you choose can depend on the field of study or how much risk you're willing to accept of incorrectly rejecting the null hypothesis. The lower the significance level, the stronger the evidence must be to reject the null hypothesis.

The critical value is a point on the test distribution that is compared to the test statistic to decide whether to reject the null hypothesis. If your test statistic is more extreme than the critical value, then you reject the null hypothesis.The critical value depends on the predetermined level of significance and the distribution that applies to the test statistic. For Z-tests, the critical values are derived from the standard normal distribution. In the exercise, the test was one-tailed and the significance level was 10%, which meant the critical value was 1.28 (looking at Z-distribution tables or using statistical software). Since our calculated Z-score was greater than the critical value, it led to the rejection of the null hypothesis and supported the alternative hypothesis.