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NSWMaths Extension 1Quick questions

Statistical Analysis (ME-S1)

Quick questions on Normal approximation of the binomial distribution: continuity, validity and z-scores

13short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is the result?
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If X∼B(n,p)X \sim B(n, p) with nn "large" (rule of thumb: both np≥5n p \ge 5 and n(1−p)≥5n (1 - p) \ge 5), then
What is when the approximation works?
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The approximation is good when:
What is continuity correction?
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The binomial is discrete; the normal is continuous. To improve the approximation, adjust the boundary by ±0.5\pm 0.5.
What is standardising?
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For a normally distributed XX with mean μ\mu and standard deviation σ\sigma, the standardised value is
What is when to use the approximation?
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For HSC Extension 1, use it when:
What is range probability?
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X∼B(50,0.4)X \sim B(50, 0.4). Approximate P(15≤X≤25)P(15 \le X \le 25).
What is probability of an exact value?
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X∼B(40,0.5)X \sim B(40, 0.5). Approximate P(X=20)P(X = 20).
What is check the conditions?
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X∼B(20,0.1)X \sim B(20, 0.1). Can we use the normal approximation?
What is at least?
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X∼B(80,0.3)X \sim B(80, 0.3). Approximate P(X≥30)P(X \ge 30).
What is direction of the continuity correction?
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P(X≤k)P(X \le k) uses k+0.5k + 0.5 in the numerator (extend the interval by half on the upper side). P(X≥k)P(X \ge k) uses k−0.5k - 0.5 (extend the interval by half on the lower side).
What is using the variance where the standard deviation is needed?
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Z=(X−μ)/σZ = (X - \mu)/\sigma uses σ\sigma, not σ2\sigma^2.
What is approximation when conditions fail?
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If npn p or n(1−p)n(1 - p) is less than 55, the normal approximation is poor.
What is forgetting which zz-table convention?
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Some tables give P(0≤Z≤z)P(0 \le Z \le z), others give P(Z≤z)P(Z \le z). Convert as needed. :::

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