Background: Between the years of 1997 and 2003, the World Health Organization collected data on head circumference, arm circumference, and other body measurements for 8440 children from Brazil, Ghana, India, Norway, Oman and the USA. All ofthe children were healthy and breastfed by mothers who did not smoke. Children in the sample lived in socioeconomic conditions favorable to normal physical growth. Variables such as parental education level and income level were used to exclude children whose growth was "environmentally constrained"; however, low-weight full term babies were included if their mothers met socioeconomic and nutritional criteria. In short, the study design attempted to include healthy children from a wide range of ethnic and cultural groups whose home environments reasonably assured the conditions for healthy physical growth.

This data was used to produce growth charts that are part of every pediatrician's toolkit for monitoring a child's overall health. Governments and the United Nations also rely on growth charts todescribe a country's overall health.

1)According tothe WHO report, girls who are one month old have a mean head circumference of36.55 centimeters with a standard deviation of1.17 cm.

a)Ann is concerned that her daughter's head is small. Her daughter has a head circumference of 33.25 centimeters when she is one-month old. If we consider measurements more than 2 standard deviations from the mean as unusual, is Ann's daughter's head measurement unusually small? Support your answer.

b)According to Medscape.com, microcephaly is a head circumference more than two standard deviations below the mean. What percentage of1-monthold girls will be categorized as having microcephaly? How do you know?

2)According to the WHO report, girls who are 2-years old have a mean head circumference of47.18 cm with a standard deviation of1.40 cm.A two-year old girl with Down's Syndrome has a head circumference of 45.2 cm. Children with Down's Syndrome typically have smaller heads, so this is not surprising.

a)Relative tothe WHO data, what is this girl's z-score? What does the z-score tell us?

b)Using the WHO data in a normal model, what percentage ofthe girls has a head circumference that is smaller than the girl with Down's Syndrome?

3)In a recent study, the percentage of students that graduate from college was collected across the U.S.Graduation rates are approximately normally distributed with mean 68.3% and standard deviation 10.2.

a)What proportion of colleges have fewer than 58.1% graduate?

b)What proportion of colleges have graduation rates between 50% and 75%?b)What proportion of colleges have graduation rates above 80%?

c)Is it unusual for a college to have a graduation rate lower than 50%?

4)Only the top 4.5% of premed students who take the MCAT are admitted to medical school in CA.Assume that scores on The MCAT scores are normally distributed with a mean =500 and standard deviation = 10.6. What is the minimum score needed to qualify for acceptance to medical school in CA?