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The sample size is a crucial component in statistical analysis, especially when determining confidence intervals. In this example, the bank manager used a sample size of 120 homeowners to estimate the average monthly mortgage paid.

A larger sample size generally leads to more accurate and reliable results. This is because a bigger sample provides more data points and better coverage of the population.

One important aspect is that increasing the sample size reduces the margin of error. This is particularly beneficial when the confidence interval is too wide. By increasing the sample size, the width of the confidence interval decreases, providing a more precise estimation of the mean.

While increasing sample size is beneficial, it is also crucial to consider the cost and feasibility involved. Gathering more data can require more resources and time.

Population standard deviation, denoted as \( \sigma \), represents the dispersion of a set of values from the mean in the entire population. In our given exercise, the population standard deviation of the mortgage payments is $215.

It serves as an input in calculating the confidence interval. The formula for the confidence interval uses not just the standard deviation but also the sample size.

A smaller standard deviation indicates that the data points tend to be very close to the mean, while a larger standard deviation shows a wide range of values.

• If you can reduce the population standard deviation, it will automatically narrow the confidence interval. However, in most cases, the population standard deviation is fixed and beyond control, especially when dealing with a large population.

Understanding standard deviation helps in interpreting how much variation exists within the population data.

The Z-distribution is a key concept when calculating confidence intervals. It's a statistical distribution that represents how a set of standardized values relate to the mean in a normal distribution.

The Z-value is derived from the Z-distribution table. It corresponds to the desired confidence level. For a 97% confidence level, the Z-value is approximately 2.33.

This value helps determine the margin of error in the confidence interval formula. The formula to compute the confidence interval is: \( \bar{x} \pm Z \left( \frac{\sigma}{\sqrt{n}} \right)\), where \( \bar{x} \) is the sample mean, \( \sigma \) is the population standard deviation, and \( n \) is the sample size.

Understanding the Z-distribution is essential for interpreting how much confidence you can have in your estimates of the population parameters.

The confidence level represents how certain you are that the true population parameter lies within the calculated confidence interval. In this exercise, a 97% confidence level is used, indicating a very high degree of confidence in the results.

The confidence level affects the width of the confidence interval through the Z-value. A higher confidence level results in a wider interval because you want to capture the true population mean with higher certainty.

• To narrow the confidence interval, you could lower the confidence level, but this comes at the cost of being less sure that the interval contains the true mean.
• Selecting the right confidence level depends on the context of the decision being made and the acceptable level of risk.

Confidence levels are crucial for quantifying the reliability of your estimation. It helps in decision-making processes, especially when you need to balance precision with certainty.