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When performing hypothesis testing, the null hypothesis is our starting point. You can think of it as a statement that there is no effect or no difference in the specific situation. In this exercise, the null hypothesis, denoted as \(H_0\), asserts that the mean number of credit cards that undergraduates typically carry is equal to 4.09. It's a claim of no change or no deviation from the established mean.
When testing hypotheses, we assume that the null hypothesis is true until we have enough evidence to prove otherwise. The burden is on us to show evidence against \(H_0\). This process requires careful calculation and comparison with statistical benchmarks. The null hypothesis is often written in the format:

• \(H_0: \mu = 4.09\)

This indicates that under the null hypothesis, the population mean is thought to be 4.09.

The alternative hypothesis, on the other hand, represents what we want to test or the outcome that indicates change. It's important because it contrasts with the null hypothesis. In this exercise, our alternative hypothesis is that the mean number of credit cards undergraduates carry is less than 4.09. We denote this hypothesis as \(H_1\).
While the null hypothesis assumes no change, \(H_1\) suggests a decline in the mean number of credit cards among students. Formally, the alternative hypothesis is expressed as:

• \(H_1: \mu < 4.09\)

The goal of hypothesis testing is to determine if there's enough statistical evidence to support the alternative over the null hypothesis. Essentially, we try to find reasons compelling enough to convince us that \(H_1\) could be more plausible than \(H_0\). This process involves statistical calculations and significance testing.

The Z-statistic is a critical part of hypothesis testing, especially when dealing with sample means. It helps us understand how far the sample mean is from the population mean — measured in standard deviation units. The Z-statistic provides a way to decide if the evidence is strong enough to reject the null hypothesis.
The formula to compute the Z-statistic is:

• \(Z = \frac{(\overline{x} - \mu)}{(\sigma/\sqrt{n})}\)

Where:

• \(\overline{x}\) is the sample mean
• \(\mu\) is the population mean under \(H_0\)
• \(\sigma\) is the sample standard deviation
• \(n\) is the sample size

In our case, substituting the given values like sample mean 2.6, population mean 4.09, sample standard deviation 1.2, and sample size 132, enables us to calculate the Z-statistic and see if it indicates a significant difference.

The confidence level in hypothesis testing reflects the degree of certainty we have in our results. It helps us understand the margin of error we're willing to accept when making a decision about the null hypothesis. The confidence level is usually expressed as a percentage, such as 95%.
A 95% confidence level means we are 95% confident our decision to reject or not reject the null hypothesis is correct. It leaves a 5% risk (known as alpha, \(\alpha\)) that we might be wrong. In this exercise, we use a one-sided test with \(\alpha = 0.05\) which relates directly to our critical Z value of -1.645.
The concept of confidence level ties directly into the decision-making process:

• If the computed Z-statistic is beyond the critical Z value derived from our confidence level, we can reject the null hypothesis.
• If not, the evidence isn't strong enough to support \(H_1\).

Confidence levels provide a statistical benchmark for testing hypotheses, ensuring decisions have an acceptable level of risk.