What is the standard deviation and what is the probability that a student scored higher than 322 if the mean test score is 312 and 16 percent of the scores are under 307?

Assuming a normal distribution:

If 16% are less than 307, then 84% are greater than 307.

Using single tailed normal tables we want the z value which gives 84%-50% = 34% = 0.34 to the left of the mean; this is z ≈ -0.995

z = (value - mean)/standard deviation

→ standard deviation = (value - mean)/z ≈ (307 - 312)/-0.995 ≈ 5.03

The z value of 322 is z ≈ (322 - 312)/5.03 ≈ 1.99

Using the single tailed tables give the value of the normal distribution for z = 1.99 as approx 0.477 to the right of the mean.

This means that 0.477 + 0.5 = 0.977 are less than 322

→ 1 - 0.977 = 0.023 = 2.3 % are greater than 322

→ probability greater than 322 is 2.3%