Explanation:
in general we would need a probabilty table or a probability calculator service on the internet.
but in this case we know.
the man value is 64. and the standard deviation is 6.
so, one standard deviation +/- interval from the mean value is 70 and 58.
and the interval of 2 standard deviations from the mean value is 76 and 52.
and 64 and 76 are exactly the limits we are asked for.
so, remember the normal distribution rule :
68% are within 1 standard deviation.
95% are within 2 standard deviations.
99.7% are within 3 standard deviations.
and 50% are below (and also above) the mean value.
so, we know, 50% of the 350 (= 175) students scored below 64.
95% of the students scored between 76 and 52. as there are 5% left, half of these 5% (= 2.5%) scored even below 52).
so, 95% + 2.5% = 97.5% of the 350 (= 341.25) students scored below 76.
therefore, the number of students that scored between 64 and 76 is
341.25 - 175 = 166.25 ≈ 166