document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1170375>Question 1170375</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 2) The paired data below consist of the test scores of 6 randomly selected students and the number of hours they studied for the test. Use a 0.05 significance level to test the claim that there is a linear correlation between hours studied and test score.<BR>\n" );
document.write( "Hours	5	10	4	6	10	9<BR>\n" );
document.write( "Scores	64	86	69	86	59	87\r<BR>\n" );
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document.write( "1) Null and Alternative Hypothesis \r<BR>\n" );
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document.write( "2) Calculator Work\r<BR>\n" );
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document.write( "3) Test Statistic, P-Value and Correlation Coefficient r, r=0.2242\r<BR>\n" );
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document.write( "4) Conclusion about the null hypothesis\r<BR>\n" );
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document.write( "Method 1:\r<BR>\n" );
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document.write( "Method 2:\r<BR>\n" );
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document.write( "5) Final conclusion that addresses the original claim\r<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 #851211 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=851211\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this problem step-by-step.\r<BR>\n" );
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document.write( "**1) Null and Alternative Hypothesis:**\r<BR>\n" );
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document.write( "* **Null Hypothesis (H0):** There is no linear correlation between hours studied and test scores (ρ = 0).<BR>\n" );
document.write( "* **Alternative Hypothesis (H1):** There is a linear correlation between hours studied and test scores (ρ ≠ 0).\r<BR>\n" );
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document.write( "**2) Calculator Work:**\r<BR>\n" );
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document.write( "You've mentioned that the correlation coefficient, r, is 0.2242. To perform the hypothesis test, we'll need to calculate the test statistic and p-value. Most calculators with statistical functions can do this. Here's a general outline of how to do this on a graphing calculator:\r<BR>\n" );
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document.write( "* Enter the hours studied data into List 1 (L1) and the test scores into List 2 (L2).<BR>\n" );
document.write( "* Perform a linear regression t-test. The calculator will provide the t-statistic and the p-value.<BR>\n" );
document.write( "* We are given r=0.2242.<BR>\n" );
document.write( "* The number of pairs n=6.\r<BR>\n" );
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document.write( "**3) Test Statistic, P-Value, and Correlation Coefficient r:**\r<BR>\n" );
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document.write( "* **Correlation Coefficient (r):** 0.2242 (given)<BR>\n" );
document.write( "* **Test Statistic (t):**<BR>\n" );
document.write( "    * $t = \frac{r \sqrt{n-2}}{\sqrt{1-r^2}}$<BR>\n" );
document.write( "    * $t = \frac{0.2242 \sqrt{6-2}}{\sqrt{1-0.2242^2}}$<BR>\n" );
document.write( "    * $t = \frac{0.2242 \sqrt{4}}{\sqrt{1-0.05026564}}$<BR>\n" );
document.write( "    * $t = \frac{0.2242 * 2}{\sqrt{0.94973436}}$<BR>\n" );
document.write( "    * $t = \frac{0.4484}{0.97454316}$<BR>\n" );
document.write( "    * t = 0.4601<BR>\n" );
document.write( "* **P-Value:**<BR>\n" );
document.write( "    * Degrees of freedom (df) = n - 2 = 6 - 2 = 4<BR>\n" );
document.write( "    * Using a t-distribution table or calculator with df = 4 and t = 0.4601, we find the two-tailed p-value.<BR>\n" );
document.write( "    * The p-value is approximately 0.667.\r<BR>\n" );
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document.write( "**4) Conclusion about the Null Hypothesis:**\r<BR>\n" );
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document.write( "* **Method 1: Comparing P-value to Significance Level:**<BR>\n" );
document.write( "    * The p-value (0.667) is greater than the significance level (0.05).<BR>\n" );
document.write( "    * Therefore, we fail to reject the null hypothesis.<BR>\n" );
document.write( "* **Method 2: Comparing Test Statistic to Critical Value:**<BR>\n" );
document.write( "    * For a two-tailed test with df = 4 and α = 0.05, the critical t-values are ±2.776.<BR>\n" );
document.write( "    * The test statistic (0.4601) is within the range of -2.776 and 2.776.<BR>\n" );
document.write( "    * Therefore, we fail to reject the null hypothesis.\r<BR>\n" );
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document.write( "**5) Final Conclusion that Addresses the Original Claim:**\r<BR>\n" );
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document.write( "* At the 0.05 significance level, there is not sufficient evidence to support the claim that there is a linear correlation between hours studied and test scores.<BR>\n" );
document.write( "* In other words, based off of the small sample size, we can not conclude that the hours studied has a linear correlation with the test scores.<BR>\n" );
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