91Ó°ÊÓ

Sample statistics are incredibly useful when we want to understand a larger population based on a smaller group.
In this exercise, we start by looking at a sample of Wired magazine subscribers.
We found that, on average, they spend 13.4 hours per week browsing the Internet. This average is called the sample mean and is denoted by \( \bar{x} \).

Additionally, we need to consider the sample's spread, measured by the standard deviation, which here is 6.8 hours.
This tells us how varied the internet usage times are within our sample group.
Finally, the sample size \( n \), which is 352 subscribers, helps us understand how reliable our sample mean might be as an estimate of the entire population's mean.

• Sample Mean: 13.4 hours per week
• Standard Deviation: 6.8 hours
• Sample Size: 352

By focusing on sample statistics, we can make well-informed assumptions about the entire population.

The standard error (SE) is a key statistic for measuring the precision of a sample mean.
The SE gives us an indication of how much our sample mean might differ from the true population mean.

To calculate the standard error, we divide the sample standard deviation by the square root of the sample size: \[ \text{SE} = \frac{s}{\sqrt{n}} \]Substitute \( s = 6.8 \) and \( n = 352 \) to get: \[ \text{SE} = \frac{6.8}{\sqrt{352}} \approx 0.3623 \]A smaller SE indicates that the sample mean is a more accurate estimate of the population mean.
In our case, an SE of approximately 0.3623 shows a reasonably precise estimate.

• Calculation: Showcases reliability of the sample mean
• Indicates precision: Smaller SE means greater precision

This statistic allows us to confidently progress toward calculating the confidence interval.

The margin of error (ME) quantifies the amount of random sampling error in our estimate of the population mean.
It helps create a range within which the true population mean is likely to fall.

We derive the margin of error by multiplying the Z-score, which reflects our desired confidence level, by the standard error:\[ \text{ME} = Z \times \text{SE} \]For a 95% confidence interval, the Z-score is approximately 1.96. Thus, using our earlier SE value of 0.3623: \[ \text{ME} = 1.96 \times 0.3623 \approx 0.7101 \]This margin of error tells us that our estimate of the mean internet usage time will typically be within 0.71 hours of the true mean.

• Purpose: Defines precision level
• Dependence: Relies on SE and Z-score
• Outcome: Helps form a confidence interval

By knowing the margin of error, we can proceed to determine the full confidence interval.

The Z-score plays a crucial role in confidence intervals.
It represents how many standard deviations an element is from the mean.

For a 95% confidence level, a Z-score of 1.96 is used. This value reflects that we are looking to have 95% of the expected sample means within our range.

• Z-score of 1.96: Matches a 95% confidence level

Z-scores are critical not just for confidence intervals, but for any analysis involving normal distribution, standardizing data comparisons, and hypothesis testing. They translate our desired confidence level into quantifiable numbers for further calculations.
Understanding the Z-score aids the calculation of the margin of error and, ultimately, the creation of a precise confidence interval.