8.7.0: Exercises

For the following exercises, assume we’re looking at results from two different standardized tests. The first, called the ABC, has mean 250 and standard deviation 50. The second, called the XYZ, has mean 80 and standard deviation 10.

Exercise \(\PageIndex{1}\)

What would be the \(z\)-score of a result of 322 on the ABC?

Exercise \(\PageIndex{2}\)

What would be the \(z\)-score of a result of 57 on the XYZ?

Exercise \(\PageIndex{3}\)

At what percentile would a result of 211 on the ABC fall? Round your answer to the nearest percentile.

Exercise \(\PageIndex{4}\)

At what percentile would a result of 94 on the XYZ fall? Round your answer to the nearest percentile.

Exercise \(\PageIndex{5}\)

What score would fall at the first quartile on the ABC? Round your answer to the nearest whole number.

Exercise \(\PageIndex{6}\)

What score would fall at the 60th percentile on the XYZ? Round your answer to the nearest whole number.

Exercise \(\PageIndex{7}\)

Which score is better: 202 on the ABC or 72 on the XYZ?

Exercise \(\PageIndex{8}\)

Which score is better: 324 on the ABC or 94 on the XYZ?

For the following exercises, recall that if we flip a coin at least 20 times, the distribution of the number of heads is approximately normal with mean equal to half the number of flips and standard deviation equal to half of the square root of the number of flips.

Exercise \(\PageIndex{9}\)

What would be the mean and standard deviation for the number of heads in 80 coin flips?

Exercise \(\PageIndex{10}\)

What would be the mean and standard deviation for the number of heads in 144 coin flips?

Exercise \(\PageIndex{11}\)

How many heads would be at the 80th percentile for 80 coin flips? Round your answer to the nearest whole number.

Exercise \(\PageIndex{12}\)

How many heads would be at the 20th percentile for 144 coin flips? Round your answer to the nearest whole number.

Exercise \(\PageIndex{13}\)

What is the \(z\)-score for 51 heads in 80 coin flips? Round your answer to the nearest hundredth.

Exercise \(\PageIndex{14}\)

What is the \(z\)-score for 87 heads in 144 coin flips? Round your answer to the nearest hundredth.

Exercise \(\PageIndex{15}\)

Which would be more unusual: flipping 51 heads in 80 tries or flipping 87 heads in 144 tries? How do you know?

For the following exercises, use the data in “Wheat,” which gives the yield of wheat (in bushels) produced over several equally sized plots. (These data were collected as part of an experiment reported in “The experimental error of field trials” by W.B. Mercer and A.D. Hall).

Exercise \(\PageIndex{16}\)

Create a histogram of Yield to check that it’s approximately normally distributed.

Exercise \(\PageIndex{17}\)

Find the mean and standard deviation of Yield. Round answers to the nearest thousandth.

Exercise \(\PageIndex{18}\)

What proportion of yields does the normal distribution predict should fall below 4.56? Round to the nearest thousandth.

Exercise \(\PageIndex{19}\)

At what percentile does the normal distribution predict a yield of 3.6 would fall? Round to the nearest whole number.

Exercise \(\PageIndex{20}\)

What yield does the normal distribution estimate would fall at the 80th percentile? Round to the nearest hundredth.

Exercise \(\PageIndex{21}\)

What yield does the normal distribution estimate would fall at the 95th percentile? Round to the nearest hundredth.

Exercise \(\PageIndex{22}\)

If one of these plots yields 2.8 bushels, would that be a good or bad result? How do you know?

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