Purpose: Confirm prerequisite skills before working with the normal distribution: discrete distributions, mean and standard deviation, fraction/decimal arithmetic.
Prerequisite Knowledge
Question 1 [2 marks]
For \( X \sim \mathrm{Binomial}(20, 0.5) \), find \( E(X) \) and \( \mathrm{Var}(X) \). What is \( \sigma_X \)?
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Question 2 [3 marks]
A data set has mean \( \bar{x} = 50 \) and standard deviation \( s = 5 \). For each value \( x \), compute the z-score \( z = \dfrac{x - \bar{x}}{s} \):
a) \( x = 60 \) b) \( x = 47.5 \) c) \( x = 35 \)
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Question 3 [3 marks]
Sketch a bell-shaped curve labelled with mean \( \mu \) and the points \( \mu \pm \sigma \), \( \mu \pm 2\sigma \), \( \mu \pm 3\sigma \). State the empirical rule percentages (68, 95, 99.7) on the curve.
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Question 4 [3 marks]
Solve linear equations:
a) \( z = \dfrac{x - 100}{15} = 2 \), find \( x \).
b) \( z = \dfrac{50 - \mu}{4} = -1.5 \), find \( \mu \).
c) \( z = \dfrac{x - 80}{\sigma} = 1.25 \) when \( x = 90 \), find \( \sigma \).
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Question 5 [2 marks]
Convert decimals to percents and vice versa:
a) \( 0.6826 \) → percent b) \( 95\% \) → decimal c) Compute \( 1 - 0.9772 \) as a decimal and percent.
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Question 6 [3 marks]
Reflection: How would you describe a "normal" data set in your own words? Give a real-world example of measurements that you'd expect to be approximately normal (e.g., student heights, test scores).