Answer:
The estimate for the number of students who took the test is 23.
Step-by-step explanation:
The Empirical Rule states that, for a normally distributed random variable:
68% of the measures are within 1 standard deviation of the mean.
95% of the measures are within 2 standard deviation of the mean.
99.7% of the measures are within 3 standard deviations of the mean.
In this problem, we have that:
Mean = 78
Standard deviation = 6
Since the normal distribution is symmetric, 50% of the students had scores above the mean and 50% had scores below the mean.
Proportion of students who scored between 72 and 90:
72 = 78 - 6
So 72 is one standard deviation below the mean.
Of the 50% of the students who scored below the mean, 68% scored above 72.
90 = 78 + 2*6
So 90 is two standard deviations above the mean.
Of the 50% of the students who scored above the mean, 95% scored below 72.
Then
0.68*0.5 + 0.95*0.5 = 0.815
81.5% of the students scored between 72 and 90.
Nineteen students had test scores between 72 and 90. Estimate the number of students who took the test.
19 is 91.5% = 0.815 of the total number of students.
How much is 100% = 1?
19 - 0.815
x - 1
Rounding to the nearest whole number
The estimate for the number of students who took the test is 23.