If cynthia is in the 95th percentile and the mean of her test scores is 180 with a standard deviation of 15, which of the following scores would cynthia most likely get on a standardized test?
a. 205
b. 171
c. 220
d. 300
this is for apex
?

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

where: is the sample mean, 1.96 is the standard normal score associated with 95%, is the standard deviation and n is the sample size.

Since, the sample size is unknown, we reduce the formular to


Therefore, the 95% confidence interval is


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.