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Chapter 6: Problem 89

Medical school graduates who want to become doctors must pass the U.S. Medical Licensing Exam (USMLE). Scores on this exam are approximately Normal with a mean of 225 and a standard deviation of \(15 .\) Use the Empirical Rule to answer these questions. a. Roughly what percentage of USMLE scores will be between 210 and 240 ? b. Roughly what percentage of USMLE scores will be below 210 ? c. Roughly what percentage of USMLE scores will be above 255 ?

Short Answer

Expert verified
Roughly 68% of USMLE scores will be between 210 and 240, about 16% will be below 210, and around 2.5% of scores will be above 255.

Step by step solution

01

Understand the Empirical Rule

The Empirical Rule (or 68-95-99.7 rule) states that in a normal distribution, about 68% of data falls within one standard deviation (蟽) of the mean (渭), about 95% falls within two standard deviations, and about 99.7% lies within three standard deviations. This rule will be applied to answer the following questions.
02

Calculate the Range for Each Question

For each question, calculate the range in terms of standard deviations from the mean to determine which part of the rule applies. For example, 210 and 240 in question a) are 1 standard deviation below and above the mean (渭 - 蟽 = 225 - 15 = 210, 渭 + 蟽 = 225 + 15 = 240), so 68% of scores fall within this range.
03

Answer the Questions Using the Empirical Rule

For question a), since 210 and 240 are one standard deviation below and above the mean respectively, about 68% of USMLE scores will be between 210 and 240. For question b), since 210 is one standard deviation below the mean, and about 68% of scores lie within one standard deviation, that means about 32% lie outside this range. Since the scores are normally distributed, about half of this 32% (or 16%) will be below 210. For question c), 255 is two standard deviations above the mean. Approximately 95% of data lies within two standard deviations, so 5% lies outside this range. About half of 5% (or 2.5%) will be above 255.

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Most popular questions from this chapter

Quantitative SAT scores are approximately Normally distributed with a mean of 500 and a standard deviation of \(100 .\) On the horizontal axis of the graph, indicate the SAT scores that correspond with the provided \(z\) -scores. (See the labeling in Exercise 6.14.) Answer the questions using only your knowledge of the Empirical Rule and symmetry. a. Roughly what percentage of students earn quantitative SAT scores greater than \(500 ?\) i. almost all iii. \(50 \%\) \(\mathrm{v}\). about \(0 \%\) ii. \(75 \%\) iv. \(25 \%\) b. Roughly what percentage of students earn quantitative SAT scores between 400 and \(600 ?\) i. almost all iii. \(68 \%\) \(\mathrm{v}\). about \(0 \%\) ii. \(95 \%\) iv. \(34 \%\) c. Roughly what percentage of students earn quantitative SAT scores greater than \(800 ?\) i. almost all iii. \(68 \%\) \(\mathrm{v}\). about \(0 \%\) ii. \(95 \%\) iv. \(34 \%\) d. Roughly what percentage of students earn quantitative SAT scores les: than \(200 ?\) i. almost all iii. \(68 \%\) \(\mathrm{v}\). about \(0 \%\) ii. \(95 \%\) iv. \(34 \%\) e. Roughly what percentage of students earn quantitative SAT scores between 300 and \(700 ?\) i. almost all iii. \(68 \%\) v. \(2.5 \%\) ii. \(95 \%\) iv. \(34 \%\) f. Roughly what percentage of students earn quantitative SAT scores between 700 and 800 ? i. almost all iii. \(68 \%\) v. \(2.5 \%\) ii. \(95 \%\) iv. \(34 \%\)

Quantitative S \(\Lambda\) T scores are approximately Normally distributed with a mean of 500 and a standard deviation of 100 . Choose the correct StatCrunch output for finding the probability that a randomly selected person scores less than 450 on the quantitative SAT and report the probability as a percentage rounded to one decimal place.

Make a list of all possible outcomes for gender when a family has two children. Assume that the probability of having a boy is \(0.50\) and the probability of having a girl is also \(0.50\). Find the probability of each outcome in your list.

Babies weighing \(5.5\) pounds or less at birth are said to have low birth weights, which can be dangerous. Full-term birth weights for single babies (not twins or triplets or other multiple births) are Normally distributed with a mean of \(7.5\) pounds and a standard deviation of 1.1 pounds. a. For one randomly selected full-term single-birth baby, what is the probability that the birth weight is \(5.5\) pounds or less? b. For two randomly selected full-term, single-birth babies, what is the probability that both have birth weights of \(5.5\) pounds or less? c. For 200 random full-term single births, what is the approximate probability that 7 or fewer have low birth weights? d. If 200 independent full-term single-birth babies are born at a hospital, how many would you expect to have birth weights of \(5.5\) pounds or less? Round to the nearest whole number. e. What is the standard deviation for the number of babies out of 200 who weigh \(5.5\) pounds or less? Retain two decimal digits for use in part f. f. Report the birth weight for full-term single babies (with 200 births) for two standard deviations below the mean and for two standard deviations above the mean. Round both numbers to the nearest whole number. g. If there were 45 low-birth-weight full-term babies out of 200 , would you be surprised?

According to National Vital Statistics, the average length of a newborn baby is \(19.5\) inches with a standard deviation of \(0.9\) inches. The distribution of lengths is approximately Normal. Use technology or a table to answer these questions. For each include an appropriately labeled and shaded Normal curve. a. What is the probability that a newborn baby will have a length of 18 inches or less? b. What percentage of newborn babies will be longer than 20 inches? c. Baby clothes are sold in a "newborn" size that fits infants who are between 18 and 21 inches long. What percentage of newborn babies will not fit into the "newborn" size either because they are too long or too short?
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