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In hypothesis testing, the population proportion represents the fraction or percentage of the population that has a particular characteristic of interest. In the provided exercise, the population proportion is denoted by the symbol \( p \), and it is assigned a value of 0.65. This value signifies that 65% of the population possesses a certain attribute or trait. Understanding the population proportion is crucial because it acts as a benchmark against which sample proportions are compared to determine any significant differences. By knowing the population proportion, we set up our null hypothesis, which is a statement asserting that there is no effect or difference strictly due to random sampling.

The level of significance, symbolized by \( \alpha \), is a threshold that helps decide whether to reject the null hypothesis. In this exercise, \( \alpha \) is given as 0.05, which translates to a 5% risk of concluding that a difference exists when there is none. This 5% serves as a boundary for error probability. If the computed test statistics fall into a region where the probability is less than or equal to \( \alpha \), we reject the null hypothesis. This level helps control the Type I error rate, meaning it limits the chance of wrongly rejecting a true null hypothesis. Choosing \( \alpha \) impacts the sensitivity of the test

• Lower \( \alpha \): increases confidence in results but may overlook actual effects.
• Higher \( \alpha \): increases sensitivity but risks more false positives.

The standard deviation of the sampling distribution, often known as the standard error, provides insight into the variability of sample proportions. In hypothesis tests involving population proportions, calculating this standard deviation involves a specific formula that accounts for both the proportion and sample size: Plugging in the values from the exercise, where \( p = 0.65 \) and \( n = 1000 \), results in a standard deviation of approximately 0.0153. This measure helps determine how much the sample proportion might vary from the true population proportion by chance. A smaller standard error indicates that the sample proportion is more likely to be close to the population proportion.

In hypothesis testing, the critical value acts as a demarcation point, which guides decisions regarding the null hypothesis. For a given level of significance, \( \alpha \), you can find the critical value by referring to a standard normal distribution table. Critical values vary depending on the type of test:

• **Left-Tailed Test**: Here, you look up \( \alpha \) directly, as shown in the exercise where \( z = -1.645 \) for \( \alpha = 0.05 \).
• **Two-Tailed Test**: Here, you need to split \( \alpha \) in half and identify values on both sides of the distribution. For this exercise, \( z = \pm 1.96 \).
• **Right-Tailed Test**: This involves looking at \( 1-\alpha \), as seen with \( z = 1.645 \) for \( \alpha = 0.05 \).

By using these critical values, you establish rejection regions that determine if observations fall within unlikely ranges under the null hypothesis.

The Z-test is a statistical method used when the population variance is known and the sample size is large, typically greater than 30. It helps test the hypothesis about population proportions or means. In this exercise, the Z-test plays a pivotal role in examining whether the sample proportion significantly deviates from the specified population proportion. To perform a Z-test:

• Calculate the Z-score using the formula: \( Z = \frac{\bar{x} - \mu}{\sigma/\sqrt{n}} \)
• Compare the Z-score with critical values, which decide if you reject the null hypothesis or not.

The Z-test quickly provides evidence of any discordance between the sample findings and the hypothesized population parameter.