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When tackling a problem involving hypothesis testing, the null hypothesis ( H_0 ) is your starting point. It's a statement that assumes no effect or no difference in the context of your research question. In this case, the null hypothesis states that the population success probability is equal to 0.3. It's essentially the default position that there is no change or no deviation at work.

In mathematical terms:

• $H_0: p = 0.3$

This hypothesis is crucial because it provides a fixed value to test against. By starting with an assumption of no effect, you can use statistical analysis to determine if there is enough evidence to suggest otherwise. Remember, the goal in hypothesis testing is to determine whether to reject the null hypothesis based on the data analysis.

The alternative hypothesis ( H_a ) represents what you want to prove. In this problem, it claims that the population success probability is less than 0.3. It is the hypothesis you're aiming to support with your data.

Here's how it looks in mathematical notation:

• $H_a: p < 0.3$

The alternative hypothesis is positioned as the statement contrary to the null hypothesis. In our scenario, it suggests that the probability is actually lower than the null suggests. This conjecture is tested using statistical techniques to determine if there's enough evidence from the sample data to support it.

A Z-test is a statistical method used to determine if there's a significant difference between a sample statistic and a known population parameter.

In this example, we're dealing with proportions and have a large sample size (n = 1000), which makes the Z-test suitable. To calculate the Z-test statistic, use the following formula:

• \(z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}\)

Here:

• \(\hat{p}\) = sample proportion = 0.279
• \(p_0\) = population proportion under the null = 0.3
• n = sample size = 1000

Plug these values in to find:

• \(z = \frac{0.279 - 0.3}{\sqrt{\frac{0.3(1-0.3)}{1000}}}\) = -1.96

The resulting Z-statistic helps compare the sample data with the null hypothesis to decide if the null can be rejected.

The rejection region determines if the Z-statistic falls within a range that suggests the null hypothesis should be rejected. At a significance level (\alpha) of 0.05, this is crucial because it sets the criteria for decision-making.

For a one-tailed test, where \(H_a: p < 0.3\), you determine this by looking up the Z-score that corresponds to the 0.05 level in a standard normal distribution table. This score is roughly -1.645. Therefore, the rejection region is:

• z < -1.645

Given that our calculated Z-statistic was -1.96, it falls into this rejection region (since -1.96 < -1.645).

This placement in the rejection region leads to the conclusion that there is sufficient evidence to reject the null hypothesis, supporting the claim that the population success probability is less than 0.3.