document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1199558>Question 1199558</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given:\r<BR>\n" );
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document.write( "N = 18<BR>\n" );
document.write( "alpha = 0.05, 1 tail<BR>\n" );
document.write( "Null hypothesis: The coin is fair<BR>\n" );
document.write( "Alternative hypothesis: The coin is biased towards heads<BR>\n" );
document.write( "Obtained value: 11 heads\r<BR>\n" );
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document.write( "b) What is the Critical Value? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848197 by </B><A HREF=/tutors/aboutme.mpl?userid=textot><B>textot(100)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=textot><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848197\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **Finding the Critical Value**\r<BR>\n" );
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document.write( "* **Understand the Scenario:**<BR>\n" );
document.write( "    * We're dealing with a binomial distribution (coin flips).<BR>\n" );
document.write( "    * We want to find the critical value for a one-tailed test with α = 0.05. This means we're looking for the number of heads that would be considered statistically significant evidence that the coin is biased towards heads.\r<BR>\n" );
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document.write( "* **Determine the Critical Region:**<BR>\n" );
document.write( "    * Since we're testing for a bias towards heads, we're interested in the upper tail of the distribution. <BR>\n" );
document.write( "    * The critical region is the area in the upper tail of the binomial distribution that contains 5% of the probability.\r<BR>\n" );
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document.write( "* **Calculate the Critical Value:**<BR>\n" );
document.write( "    * We can use the cumulative distribution function (CDF) of the binomial distribution to find the critical value. <BR>\n" );
document.write( "    * The CDF gives the probability of getting *up to* a certain number of successes. <BR>\n" );
document.write( "    * We need to find the number of heads (k) where the probability of getting *k or fewer* heads is 0.95 (1 - α).\r<BR>\n" );
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document.write( "* **Using Python (as shown in the provided code):**\r<BR>\n" );
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document.write( "```python<BR>\n" );
document.write( "from scipy.stats import binom\r<BR>\n" );
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document.write( "n = 18<BR>\n" );
document.write( "p = 0.5  # Probability of heads for a fair coin<BR>\n" );
document.write( "alpha = 0.05\r<BR>\n" );
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document.write( "critical_value = binom.ppf(1 - alpha, n, p)<BR>\n" );
document.write( "print(f\"Critical Value: {critical_value}\") <BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "**Result:**\r<BR>\n" );
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document.write( "The critical value is 12. \r<BR>\n" );
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document.write( "**Interpretation:**\r<BR>\n" );
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document.write( "If we observe 12 or more heads in 18 coin flips, we would have sufficient evidence to reject the null hypothesis (that the coin is fair) at the 0.05 significance level and conclude that the coin is likely biased towards heads.<BR>\n" );
document.write( " </SPAN>\n" );
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