document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1174091>Question 1174091</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Data is collected from 30 male and 20 female employees of SMIU. 50% of the female scored 4 and remaining 2; 30% of the males scored 3, 30% scored 4 and remaining scored 5. Sd of females is 0.8 and that of males is .8. Design hypothesis and test it? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850708 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=850708\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this problem and design/test a hypothesis.\r<BR>\n" );
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document.write( "**1. Data Organization**\r<BR>\n" );
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document.write( "* **Females:**<BR>\n" );
document.write( "    * Total: 20<BR>\n" );
document.write( "    * 50% scored 4: 20 * 0.50 = 10 females scored 4.<BR>\n" );
document.write( "    * 50% scored 2: 20 * 0.50 = 10 females scored 2.<BR>\n" );
document.write( "    * Female scores: 10 with 4, 10 with 2.<BR>\n" );
document.write( "    * Sd of females = 0.8<BR>\n" );
document.write( "* **Males:**<BR>\n" );
document.write( "    * Total: 30<BR>\n" );
document.write( "    * 30% scored 3: 30 * 0.30 = 9 males scored 3.<BR>\n" );
document.write( "    * 30% scored 4: 30 * 0.30 = 9 males scored 4.<BR>\n" );
document.write( "    * 40% scored 5: 30 * 0.40 = 12 males scored 5.<BR>\n" );
document.write( "    * Male scores: 9 with 3, 9 with 4, 12 with 5.<BR>\n" );
document.write( "    * Sd of males = 0.8\r<BR>\n" );
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document.write( "**2. Calculate Means**\r<BR>\n" );
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document.write( "* **Female Mean (μf):**<BR>\n" );
document.write( "    * (10 * 4 + 10 * 2) / 20 = (40 + 20) / 20 = 60 / 20 = 3<BR>\n" );
document.write( "* **Male Mean (μm):**<BR>\n" );
document.write( "    * (9 * 3 + 9 * 4 + 12 * 5) / 30 = (27 + 36 + 60) / 30 = 123 / 30 = 4.1\r<BR>\n" );
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document.write( "**3. Hypothesis Design**\r<BR>\n" );
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document.write( "* We want to test if there's a significant difference between the mean scores of male and female employees.<BR>\n" );
document.write( "* **Null Hypothesis (H0):** There is no significant difference between the mean scores of male and female employees. (μm = μf)<BR>\n" );
document.write( "* **Alternative Hypothesis (H1):** There is a significant difference between the mean scores of male and female employees. (μm ≠ μf)<BR>\n" );
document.write( "* This is a two-tailed test.\r<BR>\n" );
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document.write( "**4. Test Statistic**\r<BR>\n" );
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document.write( "* We'll use a two-sample t-test because we're comparing two means. Since we have the standard deviation of the populations we can use a z test.<BR>\n" );
document.write( "* We will use the following formula.<BR>\n" );
document.write( "    * z = (μm - μf) / sqrt((σm^2/nm) + (σf^2/nf))<BR>\n" );
document.write( "    * Where:<BR>\n" );
document.write( "        * μm = 4.1 (male mean)<BR>\n" );
document.write( "        * μf = 3 (female mean)<BR>\n" );
document.write( "        * σm = 0.8 (male standard deviation)<BR>\n" );
document.write( "        * σf = 0.8 (female standard deviation)<BR>\n" );
document.write( "        * nm = 30 (number of males)<BR>\n" );
document.write( "        * nf = 20 (number of females)\r<BR>\n" );
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document.write( "**5. Calculation**\r<BR>\n" );
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document.write( "* z = (4.1 - 3) / sqrt((0.8^2/30) + (0.8^2/20))<BR>\n" );
document.write( "* z = 1.1 / sqrt((0.64/30) + (0.64/20))<BR>\n" );
document.write( "* z = 1.1 / sqrt(0.02133 + 0.032)<BR>\n" );
document.write( "* z = 1.1 / sqrt(0.05333)<BR>\n" );
document.write( "* z = 1.1 / 0.2309<BR>\n" );
document.write( "* z ≈ 4.76\r<BR>\n" );
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document.write( "**6. Determine the Critical Value and Make a Decision**\r<BR>\n" );
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document.write( "* Let's use a significance level (alpha) of 0.05.<BR>\n" );
document.write( "* For a two-tailed test at alpha = 0.05, the critical z-values are ±1.96.<BR>\n" );
document.write( "* Our calculated z-value (4.76) is far outside this range.<BR>\n" );
document.write( "* Therefore, 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 a statistically significant difference between the mean scores of male and female employees at SMIU.<BR>\n" );
document.write( "* The male employees, on average, scored higher than the female employees.<BR>\n" );
document.write( " </SPAN>\n" );
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