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A Z-score is a statistical measurement that helps us understand how far away our sample proportion is from the population proportion under the null hypothesis. It's a standardized way to see where our sample compares to what's expected. Let's say you have a sample proportion \( \hat{p} = 0.26 \) and the population proportion under the null hypothesis is \( p = 0.2 \). First, recognize that you have 1000 observations, so \( n = 1000 \).

The Z-score formula for a sample proportion is:

• \( z = \frac{(\hat{p} - p)}{\sqrt{\frac{p(1-p)}{n}}} \)

In this example, substitute \( \hat{p} = 0.26 \), \( p = 0.2 \), and \( n = 1000 \) into the formula. This simplifies to calculating how many standard deviations the observed sample proportion \( \hat{p} \) (0.26) is from the hypothesized population proportion \( p \) (0.2). The Z-score informs us if the difference is due to random chance or if it suggests an actual deviation from the hypothesized rate.

After substituting the values, you get:

• \( z = \frac{(0.26 - 0.2)}{\sqrt{\frac{0.2(1-0.2)}{1000}}} \)
• This will result in a numerical Z-score that needs further interpretation.

The p-value is a crucial component of hypothesis testing. It's the probability of obtaining a test statistic as extreme as the one observed, assuming that the null hypothesis is true. In simpler terms, it tells you how likely it is to get your observed results (or more extreme) if the null hypothesis was correct.

Once you've calculated the Z-score, you look up this value in a Z-table or use a calculator that provides the tail probabilities. The two-tailed test influences the p-value because it accounts for both tails of the distribution. This means you're considering extremes in both directions \((< \hat{p} \) and \( > \hat{p} \)).

To find the p-value for a two-tailed test:

• Determine the probability of the Z-score in each tail.
• Multiply by 2 because you consider both tails (left tail and right tail).

If your p-value is less than the predefined significance level (commonly 0.05), you reject the null hypothesis, suggesting that the observed sample provides enough evidence to state that the actual population proportion differs from the hypothesized proportion. If not, you fail to reject the null hypothesis.

A two-tailed test in hypothesis testing is used when there is potential for deviation in two directions. This is different from a one-tailed test where you're only interested in deviation in one direction. Here, you test if a sample proportion is either significantly greater than or significantly less than a specified population proportion.

In the context of this example, the null hypothesis \(H_0: p = 0.2\) is tested against the alternative hypothesis \(H_a: p eq 0.2\). This indicates that you're concerned with deviations on both the higher and lower sides of the population proportion. You want to know if the sample proportion \( \hat{p} = 0.26 \) deviates significantly in either direction from 0.2.

The two-tailed test requires evaluating both sides of the distribution curve. This involves determining the significance of deviations on either side of the hypothesized proportion by using distribution tables or specific software to calculate the corresponding p-values. Those p-values are then used to help you draw conclusions about the hypotheses. If your p-value from the two-tailed test falls below your chosen level of significance, then you reject the null hypothesis, showing evidence for a true difference between the sample and the population proportions.