Given:
Number of sample (n) = 50
mean = 300
standard deviation (s) = 47
confidence level = 95%
The margin of error (MOE) can be calculated using the formula:
[tex]\text{MOE = z }\times\text{ }\frac{s}{\sqrt[]{n}}[/tex]
Where z is the z-score at the given confidence level
At 95% confidence level, the z-score is 1.960
The margin of error is thus:
[tex]\begin{gathered} \text{MOE =1.96 }\times\frac{47}{\sqrt[]{50}} \\ =\text{ 13.0277} \end{gathered}[/tex]
The formula to calculate the confidence interval is:
[tex]\begin{gathered} CI=\operatorname{mean}\pm z\frac{s}{\sqrt[]{n}}^{} \\ CI\text{ = mean }\pm\text{ margin of error} \end{gathered}[/tex]
Where :
[tex]\begin{gathered} \text{lower bound = mean - margin of error} \\ \text{upper bound = mean + margin of error} \end{gathered}[/tex]
Substituting:
[tex]\begin{gathered} \text{lower bound = }300\text{ - 13.0277} \\ =\text{ 286.9723} \\ \approx\text{ 287} \\ \text{upper bound = 300 + 13.0277} \\ =\text{ 313.0277} \\ \approx\text{ 313} \end{gathered}[/tex]
Hence, if we to randomly sample from this population 100 times. The probability of having a score between 287 and 313 is 0.95
