The given information is:
- Sample mean = 25.42
- Standard deviation = 14.25724
Part 1.
We have a confidence interval (C.I.) of 80%.
We need to find the z-score for this C.I., so, let's see a diagram:
We have a two-sided confidence interval. So, we need to find the z-score for a cumulative probability of 80+10=90%=0.9.
By using a standard normal table, we can see that z-score for 0.9 is:
By using an online calculator, we find the exact z-score is 1.282.
The confidence interval is given by the formula:
[tex]C.I=\bar{x}\pm z_c\frac{s}{\sqrt{n}}[/tex]Where x is the mean of the sample, Zc is the z-score for the given confidence level, s is the standard deviation and n is the number of elements in the sample n=50.
By replacing the known values, we obtain:
[tex]\begin{gathered} 80\%C.I.=25.42\pm1.282\frac{14.25724}{\sqrt{50}} \\ \\ C.I.=25.42\pm1.282\times2.016 \\ C.I.=25.42\pm2.585 \end{gathered}[/tex]So, the lower and upper limits are:
[tex]\begin{gathered} lower:25.42-2.585=22.835 \\ upper:25.42+2.585=28.005 \end{gathered}[/tex]The margin of error is equal to half the width of the entire confidence interval, and it is also the value we add and subtract to the mean to find the confidence interval, so the margin of error is 2.585.
