7.3 Problem Set

1) The Standard Normal Curve is defined as having a mean of 0 and a standard deviation of 1.

a) What is the z-score associated with a result at the 84th percentile?

b) What is the z-score associated with a result at the 16th percentile?

c) Find a z-score such that only 5% of the Standard Normal Curve is to the right of that z-score.

d) Find a z-score such that only 35% of the Standard Normal Curve is the left of that z-score.

e) Find the two z-scores such that the middle 50% of the Standard Normal Curve is between the two z-scores.

2) Doctors often monitor their patients blood-glucose levels. It is known that for blood-glucose levels,

a) Draw and label sketch of the normal distribution for this situation marking the mean and 1, 2, and 3 standard deviations above and below the mean.

b) It turns out that doctors consider the blood-glucose level of a patient to be normal if the level is in the middle 94% of all results. What range of blood-glucose levels constitute the middle 94% of all results?

c) Patients are considered to be high risk for diabetes if their blood-glucose test comes back in the top 1% of all results. What blood-glucose level marks the start of the top 1% of blood-glucose levels?

d) Doctors also show concern if there is too little blood-glucose in a patient's system. They will prescribe treatments to patients if their blood-glucose is in the lowest 2% of all patients. What is the blood-glucose level that marks this boundary?

3) For a given population of high school juniors and seniors, the SAT math scores are normally distributed with a mean of 500 and a standard deviation of 100. For that same population, the ACT math exam has a mean of 18 with a standard deviation of 6.

a) One school requires that students score in the top 10% on their SAT math exam for admission. What is the minimum score that a student must achieve to be considered for this school?

b) Another school requires that students score in the top 40% on their ACT math exam for admission. What is the minimum score that a student must achieve to be considered for this school?

c) One particular school likes to focus on mid-level students and so they only accept students who are in the middle 50% of all ACT math test takers. Between what two scores must a student achieve in order to be considered for acceptance into this school?

d) One student boasts that they scored at the 85th percentile on their ACT math exam. Another student brags that they scored a 620 on the SAT math exam. Who did better?

4) Many athletes train to try to be selected for the US Olympic team. Suppose for the men's 100 meters, the athletes being considered for the team have a mean time of 10.06 seconds with a standard deviation of 0.07 seconds. In the final qualifying event for the team, only the top 20% of runners will be selected. What time must a runner get to be in the top 20%?

5) A high school basketball coach notices that taller players tend to have more success on his team. As a result, the coach decides that only the tallest 25% of the boys in the 11th and 12th grades will be allowed to try out for the team this year. Suppose that the mean height of 11th and 12th grade boys is 5 feet 9 inches with a standard deviation of 2.5 inches. How tall must a player be in order to be able to try out for the team?

6) A student comes home to his parents and excitedly claims that he is in the top 90% of his class. Explain why this might not be worth getting excited about.

7) At a certain fast-food restaurant, automatic soft drink filling machines have been installed. For 20-ounce cups, the machine is set to fill up the cups with 19 ounces of soda. Unfortunately, the machine is not perfectly consistent and does not always dispense 19 ounces of soda. Suppose the amount it dispenses produces a normal distribution with a mean of 19 ounces and a standard deviation of 0.6 ounces. It turns out that the 20 ounce cup will actually hold a bit more than 20 ounces. A mathematically inclined worker notices this and starts to record what happens when the machine fills the cups. It turns out that the cups overfill 2% of the time. How much soda will the 20-ounce cup actually hold?

Review Exercises

8) Adult male American bald eagles have a mean wingspan of 79 inches with a standard deviation of 3.5 inches. What percent of these eagles have wingspans longer than 7 feet?

9) Consider the data in the table below where the number of pages is the explanatory variable.

a) Create a scatterplot for the data set. Label your axes.

b) Determine the correlation coefficient, r, for the scatterplot.

c) Give the least-squares linear regression equation. Be sure to define your variables.

d) Using your answer from part c), predict the weight of a book that has 130 pages.

e) Using your answer from part c), predict the number of pages for a book that weighs 295 grams.

10) Consider a standard set of 15 pool balls. Pool balls #1-#8 are solid and pool balls #9-#15 are striped.

a) If you randomly select one pool ball, what is the probability that it is both solid and odd?

b) If you randomly select one pool ball, what is the probability that it is either solid or odd?

c) If you randomly select two pool balls without replacement, what is the probability that they are either both solid or both striped?

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