document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1200425>Question 1200425</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A pharmaceutical firm claims that a new analgesic drug relieves mild pain under standard conditions for 3 hours, with a standard deviation 1 hour. Sixteen patients are tested under the same conditions and have an average pain relief time of 2.5 hours. Test the hypothesis that the population mean of this sample is actually 3 hours. Use ( =0.05). Try to follow all steps in hypothesis testing.<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 #848136 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=848136\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Certainly, let's perform a hypothesis test to determine if the population mean pain relief time of the new analgesic drug is actually 3 hours.\r<BR>\n" );
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document.write( "**1. Set up Hypotheses**\r<BR>\n" );
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document.write( "* **Null Hypothesis (H0):** <BR>\n" );
document.write( "    * μ = 3 <BR>\n" );
document.write( "    * The population mean pain relief time is 3 hours.\r<BR>\n" );
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document.write( "* **Alternative Hypothesis (H1):** <BR>\n" );
document.write( "    * μ ≠ 3 <BR>\n" );
document.write( "    * The population mean pain relief time is not 3 hours.\r<BR>\n" );
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document.write( "**2. Given**\r<BR>\n" );
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document.write( "* Sample mean (x̄) = 2.5 hours<BR>\n" );
document.write( "* Population standard deviation (σ) = 1 hour<BR>\n" );
document.write( "* Sample size (n) = 16 patients<BR>\n" );
document.write( "* Significance level (α) = 0.05\r<BR>\n" );
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document.write( "**3. Calculate the Test Statistic (z-score)**\r<BR>\n" );
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document.write( "Since we know the population standard deviation (σ), we can use the z-test:\r<BR>\n" );
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document.write( "* z = (x̄ - μ) / (σ / √n)<BR>\n" );
document.write( "* z = (2.5 - 3) / (1 / √16) <BR>\n" );
document.write( "* z = -0.5 / 0.25<BR>\n" );
document.write( "* z = -2\r<BR>\n" );
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document.write( "**4. Determine the Critical Value**\r<BR>\n" );
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document.write( "* This is a two-tailed test (since H1 is μ ≠ 3).<BR>\n" );
document.write( "* Find the critical z-values for α/2 = 0.05/2 = 0.025 in a standard normal distribution table.<BR>\n" );
document.write( "* The critical z-values are approximately ±1.96.\r<BR>\n" );
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document.write( "**5. Decision Rule**\r<BR>\n" );
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document.write( "* If the calculated z-score falls within the critical region (z < -1.96 or z > 1.96), reject the null hypothesis.<BR>\n" );
document.write( "* Otherwise, fail to reject the null hypothesis.\r<BR>\n" );
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document.write( "**6. Make a Decision**\r<BR>\n" );
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document.write( "* Our calculated z-score (-2) is less than the lower critical value (-1.96).<BR>\n" );
document.write( "* Therefore, we **reject the null hypothesis**.\r<BR>\n" );
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document.write( "**7. Conclusion**\r<BR>\n" );
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document.write( "* At the 0.05 significance level, there is sufficient evidence to conclude that the population mean pain relief time of the new analgesic drug is **different from 3 hours**. \r<BR>\n" );
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document.write( "**In summary:**\r<BR>\n" );
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document.write( "The sample data suggests that the average pain relief time for the 16 patients is significantly different from the claimed 3 hours. This could indicate that the drug's effectiveness might vary in the general population. \r<BR>\n" );
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document.write( "**Note:**\r<BR>\n" );
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document.write( "* This analysis assumes that the sample is normally distributed or the sample size is large enough for the Central Limit Theorem to apply. <BR>\n" );
document.write( "* Further investigation might be warranted to understand the reasons for this difference.<BR>\n" );
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