document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1177044>Question 1177044</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 12.) Find the P-value for the following values of the test statistic t, sample size n, and alternate hypothesis H1. If you use Table A.3, you may specify that P is between two values.<BR>\n" );
document.write( "d.t = 3.635, n = 4, H1: μ > μ0<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850531 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.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=850531\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's find the P-value for the given t-statistic, sample size, and alternate hypothesis.\r<BR>\n" );
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document.write( "**Understanding the Problem**\r<BR>\n" );
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document.write( "* **t-statistic (t):** 3.635<BR>\n" );
document.write( "* **Sample size (n):** 4<BR>\n" );
document.write( "* **Alternate hypothesis (H1):** μ > μ0 (right-tailed test)\r<BR>\n" );
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document.write( "**Steps to Calculate the P-value**\r<BR>\n" );
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document.write( "1. **Calculate the Degrees of Freedom (df):**<BR>\n" );
document.write( "   df = n - 1 = 4 - 1 = 3\r<BR>\n" );
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document.write( "2. **Use the t-distribution:**<BR>\n" );
document.write( "   We need to find the probability that a t-distributed random variable with 3 degrees of freedom is greater than 3.635.\r<BR>\n" );
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document.write( "3. **Use a t-table or calculator/software:**<BR>\n" );
document.write( "   * **Using a t-table:** Look up the t-value in a t-table with 3 degrees of freedom. You'll find that 3.635 falls between the values for which you'd find probabilities.<BR>\n" );
document.write( "   * **Using a calculator or software:** We can use the cumulative distribution function (CDF) of the t-distribution. The P-value is 1 - CDF(t, df).\r<BR>\n" );
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document.write( "**Calculation**\r<BR>\n" );
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document.write( "Using Python (scipy.stats):\r<BR>\n" );
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document.write( "```python<BR>\n" );
document.write( "import scipy.stats as stats\r<BR>\n" );
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document.write( "t_stat = 3.635<BR>\n" );
document.write( "n = 4<BR>\n" );
document.write( "degrees_of_freedom = n - 1\r<BR>\n" );
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document.write( "p_value = 1 - stats.t.cdf(t_stat, df=degrees_of_freedom)\r<BR>\n" );
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document.write( "print(f\"The p-value is: {p_value:.4f}\")<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "The p-value is approximately 0.0179.\r<BR>\n" );
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document.write( "**Answer**\r<BR>\n" );
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document.write( "The P-value is approximately 0.0179.<BR>\n" );
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