document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1200717>Question 1200717</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Arbitron Media Research Inc. conducted a study of the iPod listening habits of men and women. One facet of the study involved the mean listening time. It was discovered that the mean listening time for a sample of 10 men was 42 minutes per day. The standard deviation was 16 minutes per day. The mean listening time for a sample of 13 women was also 42 minutes, but the standard deviation of the sample was 15 minutes. At the 0.02 significance level, can we conclude that there is a difference in the variation in the listening times for men and women?<BR>\n" );
document.write( "Compute the p-value. (Round your answer to 4 decimal places.? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848155 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=848155\">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):** σ1² = σ2² (The variances of listening times for men and women are equal)<BR>\n" );
document.write( "* **Alternative Hypothesis (H1):** σ1² ≠ σ2² (The variances of listening times for men and women are not equal)\r<BR>\n" );
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document.write( "**2. Calculate Test Statistic (F-statistic)**\r<BR>\n" );
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document.write( "* F = s1² / s2² <BR>\n" );
document.write( "* F = 16² / 15² <BR>\n" );
document.write( "* F = 256 / 225 <BR>\n" );
document.write( "* F = 1.138\r<BR>\n" );
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document.write( "**3. Determine Degrees of Freedom**\r<BR>\n" );
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document.write( "* Degrees of freedom for the numerator (df1) = n1 - 1 = 10 - 1 = 9<BR>\n" );
document.write( "* Degrees of freedom for the denominator (df2) = n2 - 1 = 13 - 1 = 12\r<BR>\n" );
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document.write( "**4. Find the P-value**\r<BR>\n" );
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document.write( "* Using an F-distribution table or statistical software (like R or Python), find the p-value associated with the calculated F-statistic (1.138), df1 = 9, and df2 = 12. <BR>\n" );
document.write( "* **P-value ≈ 0.7724** \r<BR>\n" );
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document.write( "**5. Make a Decision**\r<BR>\n" );
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document.write( "* **Significance Level (α) = 0.02**<BR>\n" );
document.write( "* **Since the p-value (0.7724) is greater than α (0.02), we fail to reject the null hypothesis.**\r<BR>\n" );
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document.write( "**Conclusion**\r<BR>\n" );
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document.write( "* At the 0.02 significance level, there is **not enough evidence** to conclude that there is a difference in the variation in listening times for men and women.\r<BR>\n" );
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document.write( "**Note:**\r<BR>\n" );
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document.write( "* This analysis assumes that the listening times for both men and women are normally distributed. <BR>\n" );
document.write( "* If the normality assumption is not met, other tests like the Levene's test or Bartlett's test might be more appropriate.<BR>\n" );
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