document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1200870>Question 1200870</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 5.Summary is obtained from two independent Normal samples <BR>\n" );
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document.write( " 	                  Sample1 	           Sample2 <BR>\n" );
document.write( "Sample size                    20                    40<BR>\n" );
document.write( "Sample mean                    28                    30<BR>\n" );
document.write( "Sample standard deviation 	8                    10\r<BR>\n" );
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document.write( "a) Test whether one can assume equal variances. <BR>\n" );
document.write( "b) With a suitable test procedure, test for equality of mean </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848033 by </B><A HREF=/tutors/aboutme.mpl?userid=asinus><B>asinus(45)</B></A><img src=\"/images/stars/stars-07.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=asinus><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=848033\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **a) Test for Equal Variances (Levene's Test)**\r<BR>\n" );
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document.write( "* **Hypothesis:**<BR>\n" );
document.write( "    * **H0:** σ₁² = σ₂² (Variances are equal)<BR>\n" );
document.write( "    * **H1:** σ₁² ≠ σ₂² (Variances are not equal)\r<BR>\n" );
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document.write( "* **Levene's Test:** <BR>\n" );
document.write( "    * Levene's Test is used to assess the equality of variances between two groups. <BR>\n" );
document.write( "    * You would typically use statistical software (like R, Python with libraries like SciPy, or software like SPSS or Minitab) to perform this test. <BR>\n" );
document.write( "    * The software will output a test statistic (often an F-statistic) and a p-value.\r<BR>\n" );
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document.write( "* **Decision:**<BR>\n" );
document.write( "    * If the p-value from Levene's Test is greater than your chosen significance level (e.g., α = 0.05), you **fail to reject the null hypothesis**. This suggests that there is no significant evidence to conclude that the population variances are different. <BR>\n" );
document.write( "    * If the p-value is less than your chosen significance level, you **reject the null hypothesis**. This suggests that the population variances are likely different.\r<BR>\n" );
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document.write( "**b) Test for Equality of Means**\r<BR>\n" );
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document.write( "* **Based on the outcome of Levene's Test:**\r<BR>\n" );
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document.write( "    * **If variances are equal (Levene's Test not significant):**<BR>\n" );
document.write( "        * Use the **t-test for independent samples assuming equal variances**. <BR>\n" );
document.write( "        * This test assumes that the populations have equal variances.\r<BR>\n" );
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document.write( "    * **If variances are not equal (Levene's Test significant):**<BR>\n" );
document.write( "        * Use the **Welch's t-test for independent samples with unequal variances**. <BR>\n" );
document.write( "        * This test does not assume equal variances between the two groups.\r<BR>\n" );
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document.write( "* **Hypothesis:**<BR>\n" );
document.write( "    * **H0:** μ₁ = μ₂ (Population means are equal)<BR>\n" );
document.write( "    * **H1:** μ₁ ≠ μ₂ (Population means are not equal) \r<BR>\n" );
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document.write( "* **Perform the t-test:**<BR>\n" );
document.write( "    * Use the appropriate t-test (equal or unequal variances) based on the results of Levene's Test.<BR>\n" );
document.write( "    * The software will provide a t-statistic, degrees of freedom, and a p-value.\r<BR>\n" );
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document.write( "* **Decision:**<BR>\n" );
document.write( "    * If the p-value from the t-test is less than your chosen significance level (e.g., α = 0.05), you **reject the null hypothesis**. This suggests that there is a statistically significant difference between the population means.<BR>\n" );
document.write( "    * If the p-value is greater than or equal to the significance level, you **fail to reject the null hypothesis**. This suggests that there is not enough evidence to conclude that the population means are different.\r<BR>\n" );
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document.write( "**Important Considerations:**\r<BR>\n" );
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document.write( "* **Software Implementation:** Use statistical software (like R, Python, SPSS, Minitab) to perform the Levene's Test and the appropriate t-test. These software packages will provide the necessary calculations and statistical output.<BR>\n" );
document.write( "* **Assumptions:** Remember to check the assumptions of normality for both groups before conducting the t-tests. You can use graphical methods (histograms, Q-Q plots) or statistical tests (Shapiro-Wilk test) to assess normality.\r<BR>\n" );
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document.write( "By following these steps, you can determine whether there is a significant difference in the means of the two independent samples while appropriately accounting for the equality of variances.<BR>\n" );
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