One Scale for Two Different Contests

Mathematics · Probability · ages 16-17

Name ______________________ Date ____________

A value below the mean has a negative z-score.
Circle one: True False
A z-score of 0 means the value is equal to the mean.
Circle one: True False
Seedlings in a tray have mean 60 cm and standard deviation 5 cm. A seedling measures 70 cm. What is its z-score?
Answer: ______________
About what percent of values lie within one standard deviation of the mean?
- 68
- 50
- 95
- 99.7
What does a z-score of -1.5 mean?
- The value sits 1.5 standard deviations below the mean
- The value is 1.5 units below the mean
- The value sits 1.5 standard deviations above the mean
- The chance of the value is negative 1.5
Sam has z-scores of 1.2 and 1.2 from two different competitions and says he scored the same mark in both. Where is he wrong?
- the two competitions must have used the same scale for that to happen
- a z-score can never come out the same in two different competitions at once
- he should have added the two z-scores together before comparing them
- equal z-scores mean the same position in each field, not the same mark
Marks have μ = 50 and σ = 6. What mark has a z-score of 2?
Answer: ______________
Roughly what percentage of a normal distribution lies more than two standard deviations above the mean? Give the number only.
Answer: ______________
Heights are normal with μ = 168 cm and σ = 7 cm. Roughly what percentage of people are taller than 182 cm?
Answer: ______________
Why is the area under a whole normal curve equal to 1?
- because the curve is always drawn to a height of exactly one at its peak
- because the area is a share of the values, and every value is somewhere under it
- because the mean of any normal distribution always comes to 1 after standardising
- because the curve is symmetric, so its two halves must add up to one

Answer key

For grown-ups. Fold this page away before handing over the rest.

One Scale for Two Different Contests W1-mt_Cs32DT9KnA-s1

True · The top of the fraction is the value minus the mean, which comes out negative when the value is the smaller of the two. Dividing by a positive σ keeps the sign.
True · The top of the fraction is x - μ, which is zero exactly when x equals μ. Zero divided by σ is zero.
2 · (70 - 60)/5 = 2, so the value sits two standard deviations above the mean.
68 · The empirical rule puts about 68 percent of values within one standard deviation of the mean.
The value sits 1.5 standard deviations below the mean · A z-score counts standard deviations from the mean, and the minus sign points below it. Raw units never enter the reading.
equal z-scores mean the same position in each field, not the same mark · Standardising throws the scale away on purpose. With μ = 60, σ = 8 the mark would be 69.6, and with μ = 30, σ = 4 it would be 34.8, both at z = 1.2.
62 · A z of 2 means two standard deviations above the mean: 50 + 2 × 6 = 62.
2.5 · About 95 percent lies inside the two-σ band, so about 5 percent lies outside it. The curve is symmetric, so that splits into about 2.5 percent at each end.
2.5 · 182 is 14 cm above the mean, which is exactly 2σ. About 95 percent lie within two σ, so about 2.5 percent lie above the upper end.
because the area is a share of the values, and every value is somewhere under it · Area is proportion, so the total area is the proportion of values that lie somewhere, which is all of them. It is the same reason a probability distribution's entries add to 1.