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A confidence interval is a range of values, derived from the sample data, that is likely to contain the population parameter with a certain level of confidence. In our exercise, we want to determine a 95% confidence interval for the difference in mean credit limits between two different years.

To calculate this, we first need the means and standard deviations for both years. From the sample sizes and these values, we determine the standard error, which measures how much sampling variability there is in the estimate of the difference of means.

• 2008: Mean = \\(4710, Standard deviation = \\)485
• 2009: Mean = \\(4602, Standard deviation = \\)447
• Both years have a sample size of 200.

With these, we use the formula:\[ (\overline{x_1} - \overline{x_2}) \pm Z \cdot SE \]where \(\overline{x_1} - \overline{x_2}\) is the difference in sample means, \(Z\) is the z-score corresponding to the desired confidence level (1.96 for 95%), and \(SE\) is the standard error. The confidence interval gives us a range we can be 95% sure contains the true difference between the means of the two years.

The difference between two sample means can provide insights into how two populations compare. In this problem, we compare the mean credit limits from credit cards issued in 2008 versus the first part of 2009.

Calculating the difference of means involves simply subtracting the mean for 2009 from the mean for 2008:
\[\Delta \overline{x} = \overline{x_1} - \overline{x_2}\]In our scenario:

• 2008 Mean = \\(4710
• 2009 Mean = \\)4602
• Difference = \\(4710 - \\)4602 = \$108

This tells us that, on average, the credit limit was slightly higher in 2008. However, to determine if this difference is statistically significant, we also need to conduct a hypothesis test using the standard error and significance levels.

The standard error plays a crucial role in hypothesis testing and confidence intervals. It measures the dispersion or "spread" of sample means around the population mean. In other words, it indicates how much we expect a sample mean to vary from the true population mean.

In our exercise, we calculate the standard error for the difference in means using the formula:\[SE = \sqrt{\left(\frac{s_1^2}{n_1}\right) + \left(\frac{s_2^2}{n_2}\right)}\]with \(s_1\) and \(s_2\) as the sample standard deviations and \(n_1\) and \(n_2\) as the sample sizes for 2008 and 2009, respectively.

• 2008: Standard Deviation = \\(485
• 2009: Standard Deviation = \\)447
• Both Sample Sizes = 200

Calculating this gives us the standard error for the difference between these two yearly means. This helps assess the reliability of the conclusions we draw from our analyses, such as estimating confidence intervals or testing hypotheses.

The significance level is a critical concept in hypothesis testing, used to decide whether our results are statistically significant. In this exercise, we use a significance level of 2.5% (or \(\alpha = 0.025\)).

The significance level determines the threshold for rejecting the null hypothesis, often expressed as a percentage. A lower significance level means stricter criteria for confirming a statistically significant difference.
To test our hypothesis:

• Null Hypothesis \(H_0\): The mean credit limit for 2008 is not higher than that of 2009.
• Alternative Hypothesis \(H_1\): The mean credit limit for 2008 is higher than that of 2009.

We calculate the test statistic based on the difference in sample means and the corresponding standard error. Comparing it against the critical value from the Z or t-distribution (depending on sample size and standard deviation equality), we determine whether to reject \(H_0\).
Remember, if the calculated value exceeds the critical value, it suggests there's enough evidence to conclude a significant difference, thus supporting the alternative hypothesis.