Question

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
Help please?

Answer 1

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

`\bar{x}\pm1.96\left( (\sigma)/( √(n) ) \right)`
where:
`\bar{x}` is the sample mean, 1.96 is the standard normal score associated with 95%,
`\sigma` is the standard deviation and n is the sample size.

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

`\bar{x}\pm1.96(\sigma)`

Therefore, the 95% confidence interval is

`180\pm1.96(15)=180\pm29.4`

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.