به انگلیسی

One scale for two different contests

Mathematics · Probability · ages 16-17

Name ______________________ Date ____________

For a data set with positive standard deviation, a value below the mean has a negative z-score.
Circle one: True False
For a data set with positive standard deviation, a z-score of 0 means the value equals the mean.
Circle one: True False
A data set has mean 20 and standard deviation 4. The value is 18. Calculate its z-score.
Answer: ______________
For a normal model, which event is the central interval containing about 68% of the probability?
- X < μ + σ
- μ - σ < X < μ + σ
- X > μ - σ
- μ < X < μ + σ
Every value in a data set is identical, so σ = 0. Can z = (x - μ) / σ be used to assign z-scores?
- No; division by zero leaves this z-score undefined
- Yes; the zero numerator makes every z-score equal to zero
- Yes; dividing by zero makes every z-score equal to one
- No; identical values must all have negative z-scores
Sam has z = 1.2 in two competitions. He says this guarantees identical raw marks. Which explanation identifies the unsupported conclusion?
- Equal z-scores require identical means and standard deviations in both competitions
- Equal z-scores cannot occur in two competitions with different marking scales
- The two z-scores must be added before either raw mark can be compared
- Equal z-scores give equal signed σ-unit distances, but raw marks need not match
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.
-0.5 · Use forward standardisation: z = (18 - 20) / 4 = -2 / 4 = -0.5. The value is half a standard deviation below the mean.
μ - σ < X < μ + σ · The central interval extends one standard deviation below and above the mean. A single boundary or a half-interval describes a different event.
No; division by zero leaves this z-score undefined · This formula requires positive σ. With σ = 0 it involves division by zero, even though the numerator is also zero.
Equal z-scores give equal signed σ-unit distances, but raw marks need not match · Raw marks follow x = μ + zσ. For z = 1.2, μ = 60 and σ = 8 give 69.6, while μ = 30 and σ = 4 give 34.8. Equal z does not guarantee equal marks or ranks.
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.