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When dealing with hypothesis testing, we often talk about the **population mean**. This is the average of a set of values across an entire population.
In our exercise, we're investigating the mean minimum purchase amount for which Canadians use their debit cards.
The population mean is represented by the Greek letter \( \mu \). For this example, the hypothesized population mean we are testing against is \( \mu = 10 \) dollars.

• Null hypothesis (H0): The population mean is equal to a specific value (\( \mu = 10 \)).
• Alternate hypothesis (Ha): The population mean is less than the specified value (\( \mu < 10 \)).

Understanding the population mean helps us define what we expect, and this forms the backbone of our hypothesis testing.

Next, let's talk about **standard deviation**. This concept tells us how much variation or "spread" exists in the data.
In simpler terms, it describes how far the data points tend to be from the mean.
In our example, the survey reported a standard deviation of \( \sigma = 7.60 \) dollars.
This value helps us estimate how much the amounts Canadians consider for debit card purchases vary around the mean.

• If the standard deviation is low, the data points are close to the mean.
• If it's high, the data points are spread out over a larger range of values.

A standard deviation helps in calculating the z-test statistic, as it is a critical part of measuring how "unusual" or significant our sample data is compared to what we expect.

The **Z-Test** is a statistical test used to determine if there's a significant difference between a sample mean and a population mean.
For this, we use the formula:\[ z = \frac{(X_{\text{bar}} - \mu)}{(\sigma / \sqrt{n})} \]where:- \( X_{\text{bar}} \) is the sample mean (\\(9.15).- \( \mu \) is the population mean (\\)10).- \( \sigma \) is the standard deviation (\$7.60).- \( n \) is the number of observations in the sample (2000).The Z-Test calculates how many standard deviations our sample mean is from the population mean.
In our example, the z-test statistic equals \(-1.778\).
If the z-value is extreme, it suggests that it's unlikely the sample mean would occur if \( \mu = 10 \) is true.
The Z-Test is essential for testing our hypothesis because it allows us to quantify how different our sample is from what we would expect.

The **significance level** is the probability of rejecting the null hypothesis when it is actually true. It’s like drawing a line in the sand about how much evidence we need before deciding that our sample data truly reflect the broader population.
We represent this by \( \alpha \).
In our exercise, \( \alpha = 0.01 \), which signifies we are quite strict—a higher standard is needed for the data to influence our conclusion.

• A common significance level is \(0.05\), indicating a 5% risk of concluding that a difference exists when there isn't one.
• For \( \alpha = 0.01 \), we only accept a 1% chance of making an incorrect decision.

The significance level directly impacts the decision rule in hypothesis tests.
Since the p-value of 0.0378 in our example is larger than the 0.01 significance level, we "fail to reject" the null hypothesis.
This means there isn’t enough convincing evidence to suggest Canadians believe \( \mu < 10 \) for debit card use.