The Center for Disease Control has determined that one in 68 children have been
ID: 3125289 • Letter: T
Question
The Center for Disease Control has determined that one in 68 children have been identified as being affected by Broad Spectrum Disorders or Autism. A local middle school has determined that out of the 1850 students at the school, 35 students are officially documented as having a Broad Spectrum Disorder.
In determining whether this is alarming, the school desires some information. Please answer the following questions and support your answers:
A: Can you consider this a binomial experiment using the 4 criteria for binomial? Explain.
B: In a school this size, how many students would we expect to have a Broad Spectrum Disorder?
C: Assuming that this might be unusual, we wish to quantify the likelihood of having this many affected children in the population. To do this, find the probability of 35 or more students having Broad Spectrum Disorder? (If appropriate, use the normal approximation)
D: Using the information in part 3, make a statement as to how unusual you feel this situation is. In this problem, we are using data to determine likelihood. The importance of this activity is paramount to securing Federal funding for special programs. (using the 4 criteria for binomial)
Explanation / Answer
General ratio of children affected from BSD = 1/68
The ratio of children affected from BSD in the school = 35/1850 = 7/390
A: The binomial experiment has following characterstics. Checking whether this satisfies to be a binomial experiment:
Thus it is a binomial experiment.
B: Normally, considering the general ratio, the expected no. of affected children = 1/68 * 1850
= 27.2059 ~ 28 children
C: As per the study, no. of children which will have one affected child = 68
As per the school records, no. of children which will have one affected child = 1850/35 = 52.857 ~ 53
Total no of trials = 1850
Being binomial, the no of possible outcomes from 1850 trials = 2^1850
Probability of no child affected = P(X=0) = (67/68)^(1850-0) * (1/68 ^ 0)
Probability of one child affected = P(X=1) = (67/68)^(1850-1) * (1/68 ^ 1)
Similarly Probability of 34 child affected = P(X=34) = (67/68)^(1850-34) * (1/68 ^ 34)
Probability of 35 child or more affected = 1 - [P(X=0) + P(X=1) +....P(X=34)]
= 1 - 0.479253700311524 = 0.520746299688476 ~ 0.5207 = 52.07%
D: This is very unusual as this means that there will be more than 52% likelihood of having an autism affected child in 35 students.