document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1193650>Question 1193650</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Matt thinks that he has a special relationship with the number 2. In particular, Matt thinks that he would roll a 2 with a fair 6-sided die more often than you'd expect by chance alone. Suppose p is the true proportion of the time Matt will roll a 2.\r<BR>\n" );
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document.write( "Now suppose Matt makes n = 50 rolls, and a 2 comes up 11 times out of the 50 rolls. Determine the P-value of the test. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848618 by </B><A HREF=/tutors/aboutme.mpl?userid=proyaop><B>proyaop(69)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=proyaop><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=848618\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Set up Hypotheses**\r<BR>\n" );
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document.write( "* **Null Hypothesis (H0):** p = 1/6 (Matt has no special relationship with the number 2; the probability of rolling a 2 is 1/6)<BR>\n" );
document.write( "* **Alternative Hypothesis (H1):** p > 1/6 (Matt has a special relationship with the number 2; the probability of rolling a 2 is greater than 1/6)\r<BR>\n" );
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document.write( "**2. Calculate the Test Statistic**\r<BR>\n" );
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document.write( "* **Sample Proportion (p-hat):** <BR>\n" );
document.write( "    * p-hat = Number of successes / Number of trials = 11/50 = 0.22\r<BR>\n" );
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document.write( "* **Expected Number of Successes (under H0):**<BR>\n" );
document.write( "    * Expected Successes = n * p0 = 50 * (1/6) = 8.33\r<BR>\n" );
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document.write( "* **Standard Error:**<BR>\n" );
document.write( "    * Standard Error = √(p0 * (1 - p0) / n) = √((1/6) * (5/6) / 50) ≈ 0.0373\r<BR>\n" );
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document.write( "* **Test Statistic (z-score):**<BR>\n" );
document.write( "    * z = (p-hat - p0) / Standard Error = (0.22 - 1/6) / 0.0373 ≈ 3.46\r<BR>\n" );
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document.write( "**3. Determine the P-value**\r<BR>\n" );
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document.write( "* **P-value:** The probability of observing a z-score as extreme or more extreme than the calculated z-score (3.46) under the null hypothesis.<BR>\n" );
document.write( "* **Using a standard normal distribution table or statistical software:**<BR>\n" );
document.write( "    * P(Z > 3.46) ≈ 0.0003 \r<BR>\n" );
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document.write( "**Interpretation**\r<BR>\n" );
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document.write( "* The P-value of 0.0003 is very small. This means that if the null hypothesis (p = 1/6) were true, it would be extremely unlikely to observe 11 or more twos in 50 rolls.\r<BR>\n" );
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document.write( "**Conclusion**\r<BR>\n" );
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document.write( "* Since the P-value is very small (typically, a significance level of 0.05 is used), we have strong evidence to reject the null hypothesis. <BR>\n" );
document.write( "* There is strong evidence to suggest that Matt may have a special relationship with the number 2, and he rolls it more often than expected by chance.\r<BR>\n" );
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document.write( "**Note:** This analysis assumes that the rolls of the die are independent and that the die is fair except for the potential bias towards rolling a 2.<BR>\n" );
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