document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1171066>Question 1171066</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The performance of 50 Psychology students showed a mean of 98 with a standard deviation of 9.2. A sample of 40 students showed an average performance of 95 with a standard deviation of 7.3. Is there a difference in performance between the two samples using 0.05 level, two-tailed? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850994 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=850994\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to conduct the hypothesis test to determine if there's a difference in performance between the two samples.\r<BR>\n" );
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document.write( "**1. State the Hypotheses**\r<BR>\n" );
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document.write( "* **Null Hypothesis (H₀):** There is no difference in the mean performance between the two samples.<BR>\n" );
document.write( "    * H₀: μ₁ = μ₂<BR>\n" );
document.write( "* **Alternative Hypothesis (H₁):** There is a difference in the mean performance between the two samples.<BR>\n" );
document.write( "    * H₁: μ₁ ≠ μ₂ (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 independent samples and we have sample standard deviations, we will use a two-sample z-test. (We can use a z-test because the sample sizes are reasonably large.)\r<BR>\n" );
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document.write( "The formula for the z-statistic is:\r<BR>\n" );
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document.write( "z = (x̄₁ - x̄₂) / √( (σ₁²/n₁) + (σ₂²/n₂) )\r<BR>\n" );
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document.write( "Where:\r<BR>\n" );
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document.write( "* x̄₁ = mean of sample 1<BR>\n" );
document.write( "* x̄₂ = mean of sample 2<BR>\n" );
document.write( "* σ₁ = standard deviation of sample 1<BR>\n" );
document.write( "* σ₂ = standard deviation of sample 2<BR>\n" );
document.write( "* n₁ = sample size of sample 1<BR>\n" );
document.write( "* n₂ = sample size of sample 2\r<BR>\n" );
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document.write( "**3. Calculate the Test Statistic**\r<BR>\n" );
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document.write( "* x̄₁ = 98<BR>\n" );
document.write( "* σ₁ = 9.2<BR>\n" );
document.write( "* n₁ = 50<BR>\n" );
document.write( "* x̄₂ = 95<BR>\n" );
document.write( "* σ₂ = 7.3<BR>\n" );
document.write( "* n₂ = 40\r<BR>\n" );
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document.write( "z = (98 - 95) / √((9.2²/50) + (7.3²/40))<BR>\n" );
document.write( "z = 3 / √((84.64/50) + (53.29/40))<BR>\n" );
document.write( "z = 3 / √(1.6928 + 1.33225)<BR>\n" );
document.write( "z = 3 / √3.02505<BR>\n" );
document.write( "z = 3 / 1.739<BR>\n" );
document.write( "z ≈ 1.725\r<BR>\n" );
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document.write( "**4. Determine the Critical Value or P-value**\r<BR>\n" );
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document.write( "* Significance level (α) = 0.05<BR>\n" );
document.write( "* Type of test: Two-tailed\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, the critical z-values are ±z<sub>α/2</sub> = ±z<sub>0.025</sub>.<BR>\n" );
document.write( "    * Using a standard normal distribution table or a calculator, we find that z<sub>0.025</sub> ≈ 1.96.\r<BR>\n" );
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document.write( "* **P-value Approach:**<BR>\n" );
document.write( "    * Using a z-table or calculator, we find the p-value associated with z = 1.725 for a two-tailed test.<BR>\n" );
document.write( "    * P(Z > 1.725) ≈ 0.0423<BR>\n" );
document.write( "    * P(Z < -1.725) ≈ 0.0423<BR>\n" );
document.write( "    * p-value = 2 \* 0.0423 ≈ 0.0846\r<BR>\n" );
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document.write( "**5. Make a Decision**\r<BR>\n" );
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document.write( "* **Critical Value Approach:**<BR>\n" );
document.write( "    * The calculated z-statistic is 1.725.<BR>\n" );
document.write( "    * The critical z-values are ±1.96.<BR>\n" );
document.write( "    * Since |1.725| < 1.96, we fail to reject the null hypothesis.\r<BR>\n" );
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document.write( "* **P-value Approach:**<BR>\n" );
document.write( "    * The p-value is approximately 0.0846.<BR>\n" );
document.write( "    * The significance level is 0.05.<BR>\n" );
document.write( "    * Since the p-value (0.0846) is greater than the significance level (0.05), 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 sufficient evidence at the 0.05 significance level to conclude that there is a difference in performance between the two samples. </SPAN>\n" );
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