document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1163991>Question 1163991</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In a poll conducted prior to a certain election, 37% of respondents said they preferred candidate 𝐴, 33% said they preferred candidate 𝐵, and 30% said they preferred candidate 𝐶. At least how many people would have to be surveyed for you to be confident that candidate 𝐴 is truly in the lead, candidate 𝐵 is truly in second place, and candidate 𝐶 is truly in third place?\r<BR>\n" );
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document.write( "𝑛>____?\r<BR>\n" );
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document.write( "(Enter a number only.) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #854357 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2264)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=854357\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> ```python?code_reference&code_event_index=3<BR>\n" );
document.write( "import numpy as np<BR>\n" );
document.write( "from scipy.stats import norm\r<BR>\n" );
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document.write( "def find_n_multinomial_order(p, conf=0.95):<BR>\n" );
document.write( "    p_a, p_b, p_c = p<BR>\n" );
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document.write( "    # Hypothesis: pi_a > pi_b AND pi_b > pi_c<BR>\n" );
document.write( "    # We use the Wald-type approach for differences:<BR>\n" );
document.write( "    # d1 = p_a - p_b (must be > 0)<BR>\n" );
document.write( "    # d2 = p_b - p_c (must be > 0)<BR>\n" );
document.write( "    <BR>\n" );
document.write( "    # Variance of p_i - p_j is [p_i(1-p_i) + p_j(1-p_j) + 2*p_i*p_j] / n<BR>\n" );
document.write( "    # Note: cov(p_i, p_j) = -p_i * p_j / n<BR>\n" );
document.write( "    <BR>\n" );
document.write( "    z = norm.ppf(1 - (1 - conf)/2) # Standard two-tailed Z for 95% confidence<BR>\n" );
document.write( "    <BR>\n" );
document.write( "    # We want lower bound of CI for p_a - p_b to be > 0:<BR>\n" );
document.write( "    # (p_a - p_b) - z * sqrt([p_a(1-p_a) + p_b(1-p_b) + 2*p_a*p_b] / n) > 0<BR>\n" );
document.write( "    # n > z^2 * [p_a(1-p_a) + p_b(1-p_b) + 2*p_a*p_b] / (p_a - p_b)^2<BR>\n" );
document.write( "    <BR>\n" );
document.write( "    var_ab = p_a*(1-p_a) + p_b*(1-p_b) + 2*p_a*p_b<BR>\n" );
document.write( "    n1 = (z**2 * var_ab) / (p_a - p_b)**2<BR>\n" );
document.write( "    <BR>\n" );
document.write( "    # We want lower bound of CI for p_b - p_c to be > 0:<BR>\n" );
document.write( "    var_bc = p_b*(1-p_b) + p_c*(1-p_c) + 2*p_b*p_c<BR>\n" );
document.write( "    n2 = (z**2 * var_bc) / (p_b - p_c)**2<BR>\n" );
document.write( "    <BR>\n" );
document.write( "    return n1, n2\r<BR>\n" );
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document.write( "p = [0.37, 0.33, 0.30]<BR>\n" );
document.write( "n1, n2 = find_n_multinomial_order(p)<BR>\n" );
document.write( "print(f\"95% Confidence: n1={n1}, n2={n2}\")\r<BR>\n" );
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document.write( "# Let's check 90% as well<BR>\n" );
document.write( "n1_90, n2_90 = find_n_multinomial_order(p, 0.90)<BR>\n" );
document.write( "print(f\"90% Confidence: n1={n1_90}, n2={n2_90}\")\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "```text?code_stdout&code_event_index=3<BR>\n" );
document.write( "95% Confidence: n1=1676.7967752329873, n2=2685.1797156651887<BR>\n" );
document.write( "90% Confidence: n1=1180.9697177126488, n2=1891.17487441269\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "To determine the minimum sample size ($n$) required to be confident that the candidates are truly in the order $A > B > C$, we need to ensure that the differences between their support levels are **statistically significant**. \r<BR>\n" );
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document.write( "In a multinomial distribution (where respondents choose between three categories), the variance of the difference between two proportions $p_i$ and $p_j$ is given by:<BR>\n" );
document.write( "$$\text{Var}(p_i - p_j) = \frac{p_i(1-p_i) + p_j(1-p_j) + 2p_ip_j}{n}$$\r<BR>\n" );
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document.write( "To be confident in the ranking, we must satisfy two conditions simultaneously:<BR>\n" );
document.write( "1.  **Condition 1 ($A > B$):** The margin of error for the difference $p_A - p_B$ (which is $0.04$) must be small enough that the lower bound of the confidence interval is greater than $0$.<BR>\n" );
document.write( "2.  **Condition 2 ($B > C$):** The margin of error for the difference $p_B - p_C$ (which is $0.03$) must be small enough that the lower bound of the confidence interval is greater than $0$.\r<BR>\n" );
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document.write( "Using a standard **95% confidence level** ($Z \approx 1.96$):<BR>\n" );
document.write( "* For $A$ vs $B$: $n > \frac{1.96^2 \times (0.37 \cdot 0.63 + 0.33 \cdot 0.67 + 2 \cdot 0.37 \cdot 0.33)}{0.04^2} \approx 1,677$<BR>\n" );
document.write( "* For $B$ vs $C$: $n > \frac{1.96^2 \times (0.33 \cdot 0.67 + 0.30 \cdot 0.70 + 2 \cdot 0.33 \cdot 0.30)}{0.03^2} \approx 2,685$\r<BR>\n" );
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document.write( "The more restrictive condition is the smaller gap between candidate $B$ and candidate $C$. Therefore, to be confident in the entire 1-2-3 ranking, you would need a sample size larger than the highest calculated threshold.\r<BR>\n" );
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document.write( "2685 </SPAN>\n" );
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