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The standard error (SE) is an important statistic that helps you understand how much your sample mean is expected to fall from the actual population mean. It's like a measure of how much error you might have made from using a sample instead of the whole population.
To calculate the standard error, you use the formula:

• Standard Error, SE = \( \frac{s}{\sqrt{n}} \)

You divide the sample standard deviation \( s \) by the square root of the sample size \( n \). In our exercise, we had a standard deviation of 2.1 and a sample size of 70.
When you plug these numbers into the formula, it looks like this:

• SE = \( \frac{2.1}{\sqrt{70}} \)

After doing the math, the standard error is approximately 0.251. This tells you how much the mean of your sample could deviate from the true population mean. It's useful when you want to create confidence intervals and when interpreting the precision of your estimate.

The margin of error (E) signifies how much you expect your estimate of the population parameter, such as the mean, might shift from the real population value. It's a range around your sample mean where the true population mean is likely to be found.
Calculating the margin of error is straightforward once you have the standard error and the Z-score. The formula is:

• Margin of Error, E = \( Z \times SE \)

In our example, with a 95% confidence level, we use a Z-score of 1.96.
The calculation goes like this:

• E = 1.96 \times 0.251

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This results in a margin of error of approximately 0.491. This means the true mean for the population should fall within plus or minus this amount from the sample mean.

The Z-score is a statistic that helps describe how a particular data point in your sample relates to the overall distribution. In the context of confidence intervals, it helps gauge how confident you are about the sample statistics in representing the population.
A Z-score is derived from a standard normal distribution, which has a mean of 0 and a standard deviation of 1.

• If the Z-score is high, it means the data point is far from the mean.
• If the Z-score is low, the data point is close to the mean.

For constructing a confidence interval, you pick a Z-score that corresponds to your desired confidence level.
In our case, we used a Z-score of 1.96 for a 95% confidence level. This score ensures that 95% of the possible sample means would fall within this range from your population mean, making it a good measure for how far off your estimates might be.
Understanding Z-scores is vital for making confident inferences about the population from a sample.