Math Chapter 3
Give an example in which the absolute error is small but the relative error is large.
A chemist has 2.9 mg of substance, but a scale measures 2.1 mg. The absolute error is only 0.8 mg, but the relative error is approximately 28%.
Give an example in which the absolute error is large but the relative error is small.
A company projects sales of $7.30 billion and true sales turn out to be $7.32 billion.
What is an order of magnitude? Explain why such an estimate can be useful even though it may be as much as 10 times too large or too small.
An order of magnitude estimate specifies only a broad range of values, usually within one or two powers of ten. These estimates are useful because they provide a way to easily tell the difference in groups of values, such as populations.
The sales tax rate in a city is 8.7%. Find the tax charged on a purchase of $265, and the total cost.
How much tax is charged on a purchase of $265? $23.06 and the total cost is $288.06
Give an example of how "percent" (%) and "percentage points" can differ for the same situation. Choose the correct answer below.
If a savings account that previously offered 2% interest now offers 6% interest, then the percent increase is 200% and the percentage point increase is 4.
What is the difference between the terms "percent" (%) and "percentage points"?
Percent is used to describe a relative change or difference. Percentage points are used to describe an absolute change or difference.
Briefly describe scientific notation. How is it useful for writing large and small numbers? How is it useful for making approximations?
Scientific notation is a format in which a number is expressed as a number between 1 and 10 multiplied by a power of 10. It is useful for writing large and small numbers because it offers a way to save space and avoid errors in calculations. Scientific notation is useful in making approximations because it provides a quick and easy way to decide if calculated values are reasonable.
Explain how it is possible for a very accurate polygraph or drug test to result in a large proportion of false accusations. Suppose a positive result is that the test results show not lying/no drug use. Also, assume the probability of lying/drug use is very low. Choose the correct answer below.
Since the probabilities of both a false negative and a true negative are very low, the proportion of false negatives to total negatives will be large.
Find the absolute change and the percentage change in the following case. The number of daily news paper in a country was 2071 in 1900 and 1247 in 2010.
The absolute change is -824 (absolute change = new quantity - old quantity)
Distinguish between the absolute error and the relative error in a measurement.
The absolute error describes how far a measured (or claimed) value lies from the true value. The relative error compares the size of the absolute error to the true value and is often expressed as a percentage.
In the following statement, express the first number as a percentage of the second number. The full-time year-round median salary for U.S. men in 2010 was $42,800, and the full-time year-round salary for U.S. women in 2010 was $35,700.
The full-time year round median salary for U.S. men in 2010 was 119.9% of the full-time year round median salary for U.S. women in 2010.
Fill in the blank. The retail cost of a TV is 50% more than its wholesale cost. Therefore, the retail cost is ____ times the wholesale cost.
The retail cost is 1.5 times the wholesale cost.
Decide whether the following statement makes sense (or is clearly true) or does not make sense (or is clearly false). Explain your reasoning. If John earns 20% more than Mary does, then Mary must earn 20% less than John does.
The statement does not make sense because if John earns 20% more than Mary, then Mary must earn approximately 16.7% less than John does.
Decide whether the following statement makes sense (or is clearly true) or does not make sense (or is clearly false). Explain your reasoning. My older child weighs 25% more than my younger child.
The statement makes sense because if the older child weighs 25% more than the younger child, then his weight is 1.25 times the younger child's weight, which is possible.
Decide whether the following statement makes sense (or is clearly true) or does not make sense (or is clearly false). Explain your reasoning. The rate of return on our fund increased by 50%, to 15%.
The statement makes sense because if the rate of return on our fund increased by 50%, to 15%, then the previous rate is 10%, which makes sense.
Decide whether the following statement makes sense (or is clearly true) or does not make sense (or is clearly false). Explain your reasoning. We found that these rare cancers were 700% more common in children living near the toxic landfill than in the general population.
The statement makes sense because the number of rare cancers found in children living near the toxic landfill is 8 times the number of rare cancers found in the general population, which is possible.
Does this statement make sense? "In many European countries, the percentage change in population has been negative in recent decades."
The statement makes sense. A percentage change in population can be negative if the data reflects that.
How do you find relative change?
absolute change / old quantity x 100
What is Simpson's paradox?
it is possible for a set of data to give different results in each of several groups than it does when the groups are taken together
