document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1177450>Question 1177450</A>: <I><SPAN STYLE=\"word-break: break-all;\"> n a study of factors affecting hypnotism, visual analogue scale (VAS) sensory ratings were obtained for 16 subjects. For these sample ratings, the mean is 8.33 with a standard deviation of 1.96. At 05.0level of significance test the claim that this sample comes from a population with a mean rating of less than 9.00.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850457 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</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=850457\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. State the Hypotheses**\r<BR>\n" );
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document.write( "* Null Hypothesis (H0): The population mean rating is equal to 9.00 (µ = 9.00)<BR>\n" );
document.write( "* Alternative Hypothesis (H1): The population mean rating is less than 9.00 (µ < 9.00)\r<BR>\n" );
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document.write( "**2. Determine the Test Statistic**\r<BR>\n" );
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document.write( "Since the population standard deviation is unknown, we'll use a t-test. The test statistic is calculated as:\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "t = (x̄ - µ) / (s / √n)<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "Where:\r<BR>\n" );
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document.write( "* x̄ = sample mean (8.33)<BR>\n" );
document.write( "* µ = hypothesized population mean (9.00)<BR>\n" );
document.write( "* s = sample standard deviation (1.96)<BR>\n" );
document.write( "* n = sample size (16)\r<BR>\n" );
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document.write( "Plugging in the values:\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "t = (8.33 - 9.00) / (1.96 / √16) <BR>\n" );
document.write( "t ≈ -1.36<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "**3. Find the P-value**\r<BR>\n" );
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document.write( "We need to find the P-value associated with this t-statistic.  Since the alternative hypothesis is µ < 9.00 (a one-tailed test), we want the area to the left of t = -1.36 on a t-distribution with 15 degrees of freedom (df = n - 1 = 15).\r<BR>\n" );
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document.write( "Using a t-table or calculator, the P-value is approximately 0.096.\r<BR>\n" );
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document.write( "**4. Compare P-value to Significance Level (α)**\r<BR>\n" );
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document.write( "* α = 0.05 (given)<BR>\n" );
document.write( "* P-value (0.096) > α (0.05)\r<BR>\n" );
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document.write( "**5. Decision**\r<BR>\n" );
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document.write( "Since the P-value is greater than α, we fail to reject the null hypothesis.\r<BR>\n" );
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document.write( "**6. Conclusion**\r<BR>\n" );
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document.write( "There is not enough evidence at the 0.05 level of significance to support the claim that the sample comes from a population with a mean rating of less than 9.00.\r<BR>\n" );
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document.write( "**Interpretation**\r<BR>\n" );
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document.write( "While the sample mean (8.33) is less than 9.00, the difference is not statistically significant. The observed difference could be due to random chance. </SPAN>\n" );
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