Question

It is generally believed that nearsightedness affects about 15% of children. A school district gives vision tests to 111 incoming kindergarten children. Use the empirical rule (68%-95%-99.7% Rule) to determine what proportion of nearsighted children we might expect to see in samples of 111 children (I'm not looking for the number of children). Assume conditions are met! No need to draw the model, though if you find it easier to do so you may draw it by hand and attach a picture of your model using the picture button.

Answer 1

Solution:

Given in the question
P(Nearsightedness affects) = 0.15
Total no. of Children = 111
According to the empirical rule about 68% of the data are b/w +/-1 standard deviation from the mean so proportion are
Upper bound = 0.15+1*Sqrt(0.15*0.85/111) = 0.15 + 0.0339 = 0.1839
Lower bound = 0.15-1*sqrt(0.15*0.85/111) = 0.15-0.0339 = 0.1161
According to the empirical rule about 95% of the data are b/w +/-2 standard deviation from the mean so proportion are
Upper bound = 0.15+2*Sqrt(0.15*0.85/111) = 0.15 + 2*0.0339 = 0.2179
Lower bound = 0.15-2*sqrt(0.15*0.85/111) = 0.15-2*0.0339 = 0.0822
According to the empirical rule about 99.7% of the data are b/w +/-3 standard deviation from the mean so proportion are
Upper bound = 0.15+3*Sqrt(0.15*0.85/111) = 0.15 + 3*0.0339 = 0.2517
Lower bound = 0.15-3*sqrt(0.15*0.85/111) = 0.15 - 3*0.0339 = 0.0483