document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1171313>Question 1171313</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Inference (Two Populations), Chi-Squared Tests<BR>\n" );
document.write( "1. A study comparing children’s reading age (in months) was developed using identical twin<BR>\n" );
document.write( "toddlers. One set of 6 twins played for 2 hours each day with educational toys<BR>\n" );
document.write( "(experimental group), the corresponding set of 6 twins played for 2 hours each day with<BR>\n" );
document.write( "non-educational toys (control group). The mean difference in reading age between the<BR>\n" );
document.write( "experimental group and control group of 6 sets of twins was -2.33 months and the<BR>\n" );
document.write( "standard deviation of the sample difference was 2.16 months. Set up a hypothesis test to<BR>\n" );
document.write( "determine if there is a difference between the 2 groups and use the appropriate sample<BR>\n" );
document.write( "test statistic to determine if the difference in reading age is significant at the 5% level </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850969 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=850969\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Absolutely! Let's break down this hypothesis test step-by-step.\r<BR>\n" );
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document.write( "**1. Set up the Hypotheses:**\r<BR>\n" );
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document.write( "* **Null Hypothesis (H₀):** There is no difference in the mean reading age between the experimental group (educational toys) and the control group (non-educational toys). In other words, the mean difference is zero.<BR>\n" );
document.write( "    * H₀: μ<sub>d</sub> = 0 (where μ<sub>d</sub> is the mean difference)<BR>\n" );
document.write( "* **Alternative Hypothesis (H₁):** There is a difference in the mean reading age between the two groups.<BR>\n" );
document.write( "    * H₁: μ<sub>d</sub> ≠ 0 (two-tailed test)\r<BR>\n" );
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document.write( "**2. Determine the Test Statistic:**\r<BR>\n" );
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document.write( "Since we are comparing the means of two related groups (identical twins) and we have a small sample size (n = 6), we will use a paired t-test.\r<BR>\n" );
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document.write( "The formula for the t-statistic is:\r<BR>\n" );
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document.write( "t = (mean difference) / (standard deviation of the difference / √sample size)\r<BR>\n" );
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document.write( "t = μ<sub>d</sub> / (s<sub>d</sub> / √n)\r<BR>\n" );
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document.write( "Where:\r<BR>\n" );
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document.write( "* μ<sub>d</sub> = -2.33 months (mean difference)<BR>\n" );
document.write( "* s<sub>d</sub> = 2.16 months (standard deviation of the difference)<BR>\n" );
document.write( "* n = 6 (number of twin pairs)\r<BR>\n" );
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document.write( "**3. Calculate the Test Statistic:**\r<BR>\n" );
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document.write( "t = -2.33 / (2.16 / √6)<BR>\n" );
document.write( "t = -2.33 / (2.16 / 2.449)<BR>\n" );
document.write( "t = -2.33 / 0.882<BR>\n" );
document.write( "t ≈ -2.642\r<BR>\n" );
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document.write( "**4. Determine the Degrees of Freedom:**\r<BR>\n" );
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document.write( "Degrees of freedom (df) = n - 1<BR>\n" );
document.write( "df = 6 - 1 = 5\r<BR>\n" );
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document.write( "**5. Determine the Critical Value or P-value:**\r<BR>\n" );
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document.write( "* **P-value Approach:**<BR>\n" );
document.write( "    * Using a t-distribution table or a calculator, we find the p-value associated with a t-statistic of -2.642 and 5 degrees of freedom.<BR>\n" );
document.write( "    * Since this is a two-tailed test, we look for the probability of observing a t-statistic as extreme as -2.642 or 2.642.<BR>\n" );
document.write( "    * The p-value is approximately 0.0459.\r<BR>\n" );
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document.write( "* **Critical Value Approach:**<BR>\n" );
document.write( "    * For a two-tailed test with α = 0.05 and df = 5, the critical t-values are ±2.571.\r<BR>\n" );
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document.write( "**6. Make a Decision:**\r<BR>\n" );
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document.write( "* **P-value Approach:**<BR>\n" );
document.write( "    * Since the p-value (0.0459) is less than the significance level (0.05), we reject the null hypothesis.\r<BR>\n" );
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document.write( "* **Critical Value Approach:**<BR>\n" );
document.write( "    * Since the calculated t-statistic (-2.642) is less than the critical value (-2.571), 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( "There is sufficient evidence to conclude that there is a statistically significant difference in the mean reading age between children who play with educational toys and those who play with non-educational toys at the 5% significance level.<BR>\n" );
document.write( " </SPAN>\n" );
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