This is a confidence interval problem where we are looking for the number from the given options which falls inside the upper 95% confidence interval.
A confidence interval is a range of values so defined that there is a specified probability that the value of a parameter lies within it.
Here, we are to find a range of values so defined that there is a 95% probability that the value of Cynthia's standardized test score lies within it.
The 95% confidence interval of the mean is given by
[tex]\bar{x}\pm1.96\left( \frac{\sigma}{ \sqrt{n} } \right)[/tex]
where: [tex]\bar{x}[/tex] is the sample mean, 1.96 is the standard normal score associated with 95%, [tex]\sigma[/tex] is the standard deviation and n is the sample size.
Since, the sample size is unknown, we reduce the formular to
[tex]\bar{x}\pm1.96(\sigma)[/tex]
Therefore, the 95% confidence interval is
[tex]180\pm1.96(15)=180\pm29.4[/tex]
Now, since we are interested with the upper confidence, we have that the value of Cynthia's test score is between 180 and 180 + 29.4, i.e. between 180 and 209.4
Therefore, from the options given the score that Cynthia would most likely get on a standardized test is 205.
