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Statistical significance is a concept used in hypothesis testing to determine if the results from a study are likely to be true or occurred by chance. When we conduct a hypothesis test, we're generally trying to assess whether a sample's results reflect a real effect in the larger population.

In the context of the exercise, the researchers set a significance level of 5%. This level is also called the alpha level and represents the probability of rejecting the null hypothesis when it is actually true. By setting the significance level at 5%, the researchers accepted a 5% risk of concluding that a higher proportion of adults are not using cash when, in fact, the proportion might be at 40%.

• Significance Level: 5% (伪 = 0.05)
• Null Hypothesis (H鈧�): The true population proportion is 40%.
• Alternate Hypothesis (H鈧�): More than 40% of adults do not use cash.

When the result falls in the rejection area, it is deemed statistically significant, meaning our sample gives enough evidence to move away from the null hypothesis. Thus, in our example, rejecting the null hypothesis at this level suggests the proportion is statistically greater than the assumed 40%.

Proportion testing is a statistical method to determine whether the proportion of a certain characteristic within a population is equal to a certain value. It often uses a null hypothesis to test against an alternative hypothesis.

In our problem, the researchers wanted to test the hypothesis that more than 40% of adults go a week without using cash. The sample provided a proportion of 43%, above the claimed population proportion of 40%. By comparing the sample's proportion to the hypothesized population proportion, we determine if the observed data supports the alternative hypothesis.

• Sample Proportion: 43% (p = 0.43)
• Hypothesized Proportion: 40% (p鈧� = 0.40)

The key part of testing for a proportion is calculating the standard error (SE). This represents how much we expect the sample proportion to vary from the population proportion due to chance. In this case, the SE was given as 0.016. This standard error helped in calculating the Z-score, for determining how far the observed proportion is from the hypothesized proportion in terms of standard errors.

A Z-score is a measure that describes a value's position in relation to the mean of a group of values. It is expressed in terms of standard deviations from the mean. A Z-score summarizes how far away a particular point is from the mean in standard error units.

In our scenario, the Z-score indicates how many standard errors the sample proportion of 43% is from the hypothesized value of 40%. The formula used is:\[ Z = \frac{(p - p_0)}{SE} \]Here, \( p = 0.43 \) is the sample proportion, \( p_0 = 0.40 \) is the hypothesized population proportion, and \( SE = 0.016 \) is the standard error.

The calculated Z-score was 1.875, which is greater than the critical Z-value of 1.645 for a 5% significance level. This tells us that our sample proportion is significantly greater than the hypothesized proportion. A higher Z-score in the rejection region suggests that the difference we observe is not just due to random variation鈥攊t's likely due to a true difference in proportions. Thus, statistical methods affirm evidence against the null hypothesis in favor of the alternative.