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Math 117 – THEME PROBLEMS – the solutions to these problems are in my website Chapter 6.2 – Normal Distributions – Finding proportions (probabilities) and scores HEAD CIRCUMFERENCE 1) Past results from the National Health Survey suggest that the head circumferences of 2-month-old girls have a mean of 40.05 cm and a standard deviation of 1.64 cm. Assume the distribution is normally distributed. a) In a normal distribution usual values lie within two standard deviations of the mean. Find the minimum and maximum “usual” head circumferences. (These results could be used by a physician who can identify “unusual” circumferences that might be the result of a disorder such as hydrocephalus. Also, a small head circumference may mean that the baby is at greater risk for mental retardation or other developmental delays.) b) Determine whether a circumference of 42.6 cm would be considered “unusual”. c) Find the proportion of girls with head circumference lower than 39.5 cm. (This is the same as answering the question: If one girl is selected at random, what is the probability that her head circumference is lower than 39.5 cm? d) A head circumference of 38 cm falls in what percentile? e) In groups of 1000 girls, how many girls have 38 cm or less? f) What is the third quartile of the distribution of head circumferences of two-month old baby girls? SYSTOLIC BLOOD PRESSURE 2) For 18-24 year old women, systolic blood pressures (in mm Hg) are normally distributed with a mean of 114.8 mm Hg and a standard deviation of 13.1 mm Hg. a) Identify the population and the variable. b) Identify usual and unusual systolic blood pressures for this population according to the range rule of thumb. c) If one woman from this age group is randomly selected, what is the probability that her systolic blood pressure is (i) Between 94 and 142 mm Hg? (ii) At most 82 mm Hg? (iii) At least 143 mm Hg? d) Find the two blood pressures having these properties: the mean is midway between them and 90% of all blood pressures are between them. SAT VERBAL SCORES 3) According to the College Board, the mean SAT verbal score of students whose first language is English is 515. Assume SAT verbal scores are normally distributed with a population standard deviation of 112. a) Describe in words the population and the variable under consideration. b) Usual scores are scores within two standard deviations from the mean. Identify usual and unusual scores. c) From the point of view of probability, we say that unusual scores are scores for which the probability of observing that score or a more extreme one is less than or equal to 0.05 What is the probability that a student selected at random has an SAT verbal score of (i) At most 458? Is this an unusually low score? (In a normal distribution, “≤”, is the same as “<”) (ii) At least 745? Is this an unusually high score? (In a normal distribution, “≥”, is the same as “>”) (iii) Between 500 and 600? d) Lisa scored on the 60th percentile. What was her score? e) A college in the Northeast accepts applicants who scored in the top 5%. What is the lowest SAT verbal score accepted? f) Find the score that separates the bottom 5% of the distribution.