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When performing statistical tests, the sample size plays a crucial role in determining the reliability and validity of the results. A larger sample size generally provides more accurate insights into the population. In the exercise, we analyzed two scenarios: a sample size of 20 and a sample size of 500.

When the sample size is small, like 20, the results can be misleading because they might not represent the entire population well. This can lead to a "lack of statistical significance," meaning any observed differences could just be due to sampling variability rather than actual differences.

On the other hand, a larger sample size, like 500, captures more variability in the data, resulting in more robust results. With larger samples, even small effects can become noticeable because the data more accurately represents the population. Comparing the same difference in proportions with a sample size of 500, we suddenly find statistically significant results.

In statistics, we often compare the difference between proportions to understand if there is a meaningful distinction between two groups. Here, we found that 60% of women MBA students and 50% of men MBA students believe executives are paid too much.

This leads to a difference in proportions of 10%. While a numerical difference exists, it does not automatically imply a statistically significant difference. Observed differences can be due to random chance, especially when dealing with small sample sizes. Statistical tests help us determine if this difference is significant enough to generalize beyond our sample.

By conducting a statistical test, we can decide if the difference reflects a true effect or is merely a sampling fluctuation. Therefore, calculating the difference in proportions is a vital first step, but interpreting its significance requires more nuanced statistical methods.

The Z-score measures how many standard deviations an element is from the mean of the sample or population. In the context of comparing proportions, the Z-score helps assess whether the observed difference between two groups is meaningful.

For the exercise, the formula used to calculate the Z-score was \[ z = \frac{(p_1 - p_2) - 0}{\sqrt {p(1-p)(\frac{1}{n_1} + \frac{1}{n_2})}} \]where \( p_1 \) and \( p_2 \) are the sample proportions of women and men, and \( n_1 \), \( n_2 \) are their respective sample sizes.

In step 2 with a sample size of 20, the Z-score was approximately -1.22, indicating that the difference is not significant at the 0.05 level. However, with a sample size of 500, the Z-score was -2.83, suggesting the difference is significant. A higher Z-score indicates a greater likelihood that the observed difference is not due to random chance.

The level of significance, often denoted by \( \alpha \), is a threshold used to determine whether a hypothesis test's results are statistically significant. It represents the probability of rejecting the null hypothesis when it is true.

In the given exercise, a 0.05 level of significance was used. This means that there is a 5% chance of concluding that a difference exists when there is none. In statistical terms, if our calculated Z-score falls beyond the critical values (about -1.96 and 1.96 in standard normal distribution for two-tailed tests at \( \alpha = 0.05 \)), the result is considered statistically significant.

In step 2, with sample size 20, the Z-score did not exceed the critical range, leading us to conclude no statistical significance. However, step 3, with a larger sample size of 500, produced a Z-score beyond the critical values, indicating statistical significance. Hence, the level of significance helps in making informed decisions about the data analysis outcomes.