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When comparing proportions, we aim to determine if there is a significant difference between the fractions of a particular outcome in different groups. In the context of the problem, we have two machines, each producing a certain number of defective parts. By calculating the proportion of defective parts from each machine, we can analyze the extent to which one machine might be performing differently in terms of quality control.

To find this proportion, divide the number of defective instances by the total number of instances examined. For instance, if Machine 1 produced 12 defective parts out of 150, the proportion is calculated as \( p_1 = \frac{12}{150} = 0.08 \). Similarly, Machine 2's proportion is \( p_2 = \frac{6}{150} = 0.04 \). These proportions give us a clear numerical expression of the defect rate for each machine, which can then be directly compared.

The Standard Error (SE) plays a crucial role when evaluating the variability of sample proportions. It's essentially a measure that tells us how much the sample proportion is expected to vary from the true population proportion.

To calculate the SE of the difference between two proportions, use the formula:

• First, calculate the variance for each proportion. The formula for this is \( p(1-p)/n \) for each machine, where \( p \) is the proportion and \( n \) is the sample size.
• For Machine 1, it becomes \( p_1(1 - p_1)/n_1 = 0.08 \times 0.92/150 \).
• For Machine 2, \( p_2(1 - p_2)/n_2 = 0.04 \times 0.96/150 \).
• Next, sum these two results and take the square root: \( SE = \sqrt{\frac{0.08 \times 0.92}{150} + \frac{0.04 \times 0.96}{150}} \).

This provides the value of SE, which in this problem is 0.047. This value indicates the standard deviation of the sampling distribution of the difference in proportions.

The Z-value represents the number of standard deviations a data point is from the mean, and it is essential when constructing confidence intervals. Each confidence level corresponds to a specific Z-value that helps determine the range in which we expect our true proportion difference to lie.

A common Z-value for a 90% confidence level is 1.645. This means that the true difference in proportions lies within plus or minus 1.645 standard errors from the calculated difference, approximately 90% of the time.

Using the Z-value helps frame our results within a statistically accepted margin of error, allowing us to assert that the interval derived is statistically significant at the chosen confidence level. It’s calculated based on the standard normal distribution, a foundational concept in inferential statistics.

The Difference in Proportions allows us to quantify the disparity between two groups—in this case, the defect rates of the two machines. To find this difference, subtract the smaller proportion from the larger one: \( 0.08 - 0.04 = 0.04 \).

This 0.04 points out that there is a 4% higher defect rate in Machine 1's output compared to Machine 2. This numeric difference is the basis of determining whether observed discrepancies are significant.

To test this significance, we use the previous calculations to build a confidence interval. This interval gives a range in which we believe the true difference in defect rates falls, considering the calculated standard error and Z-value. In this exercise, the confidence interval is \((-0.037, 0.117)\). This indicates that while the observed difference is 4%, the actual difference, accounting for possible sampling error, could be slightly negative up to 11.7% higher, given the data and confidence level.