Confidence Interval Margin of Error Calculation

The question asks for the margin of error component of the 95% confidence interval for the mean. We are given:

• Sample size, $n = 16$
• Sample mean, $\bar{x} = 40$
• Standard deviation, $sd = 4.0$
• Z-score for 95% confidence, $Z = 1.96$

Calculating the Margin of Error (ME)

The formula to calculate the margin of error for a population mean using a Z-distribution is:

$ ME = Z \times \frac{sd}{\sqrt{n}} $

Substitute the given values into the formula:

1. Calculate the standard error of the mean (SEM):

   $ SEM = \frac{sd}{\sqrt{n}} = \frac{4.0}{\sqrt{16}} = \frac{4.0}{4} = 1.0 $

2. Calculate the margin of error:

   $ ME = Z \times SEM = 1.96 \times 1.0 = 1.96 $

The margin of error is $1.96$. Therefore, the 95% confidence interval is typically expressed as $\bar{x} \pm ME$, which would be $40 \pm 1.96$. The value representing the interval's range around the mean is $\pm 1.96$.