document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1178054>Question 1178054</A>: <I><SPAN STYLE=\"word-break: break-all;\"> two-tailed<BR>\n" );
document.write( "right-tailed\r<BR>\n" );
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document.write( "Based on a sample of 40 men, 40% owned cats<BR>\n" );
document.write( "Based on a sample of 40 women, 60% owned cats The test statistic is: Correct (to 2 decimals) The critical value is:  (to 2 decimals) Based on this we:<BR>\n" );
document.write( "Reject the null hypothesis<BR>\n" );
document.write( "Fail to reject the null hypothesis<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 #850371 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=850371\">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. Define the Variables**\r<BR>\n" );
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document.write( "* **p1:** Proportion of men who own cats (sample proportion p̂1 = 0.40)<BR>\n" );
document.write( "* **p2:** Proportion of women who own cats (sample proportion p̂2 = 0.60)<BR>\n" );
document.write( "* **n1:** Sample size of men (n1 = 40)<BR>\n" );
document.write( "* **n2:** Sample size of women (n2 = 40)\r<BR>\n" );
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document.write( "**2. State the Hypotheses**\r<BR>\n" );
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document.write( "* **Null Hypothesis (H0):** p1 = p2 (There is no difference in the proportion of men and women who own cats)<BR>\n" );
document.write( "* **Alternative Hypothesis (H1):** p1 ≠ p2 (There is a difference in the proportion of men and women who own cats)\r<BR>\n" );
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document.write( "Since the alternative hypothesis is \"not equal to,\" this is a **two-tailed** test.\r<BR>\n" );
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document.write( "**3. Calculate the Test Statistic (z-score)**\r<BR>\n" );
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document.write( "* **Pooled Proportion (p̂):**<BR>\n" );
document.write( "    * p̂ = (x1 + x2) / (n1 + n2) = (n1 * p̂1 + n2 * p̂2) / (n1 + n2)<BR>\n" );
document.write( "    * p̂ = (40 * 0.40 + 40 * 0.60) / (40 + 40) = (16 + 24) / 80 = 40 / 80 = 0.50<BR>\n" );
document.write( "* **Standard Error (SE):**<BR>\n" );
document.write( "    * SE = √[p̂(1 - p̂) * (1/n1 + 1/n2)]<BR>\n" );
document.write( "    * SE = √[0.50 * 0.50 * (1/40 + 1/40)] = √[0.25 * (2/40)] = √[0.25 * 0.05] = √0.0125 ≈ 0.1118<BR>\n" );
document.write( "* **Test Statistic (z):**<BR>\n" );
document.write( "    * z = (p̂1 - p̂2) / SE<BR>\n" );
document.write( "    * z = (0.40 - 0.60) / 0.1118 = -0.20 / 0.1118 ≈ -1.789\r<BR>\n" );
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document.write( "Rounded to two decimal places, the test statistic is **-1.79**.\r<BR>\n" );
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document.write( "**4. Determine the Critical Value**\r<BR>\n" );
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document.write( "* **Significance Level (α):** We need a significance level to determine the critical value. Let's assume a common significance level of α = 0.05.<BR>\n" );
document.write( "* **Two-Tailed Test:** For a two-tailed test with α = 0.05, the critical values are ±z(α/2) = ±z(0.025).<BR>\n" );
document.write( "* **Critical Value:** From the standard normal distribution table, z(0.025) ≈ 1.96. Therefore, the critical values are ±1.96.\r<BR>\n" );
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document.write( "**5. Make a Decision**\r<BR>\n" );
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document.write( "* **Compare Test Statistic and Critical Value:**<BR>\n" );
document.write( "    * The test statistic (-1.79) is within the range of the critical values (-1.96 and 1.96).<BR>\n" );
document.write( "    * |-1.79| < 1.96\r<BR>\n" );
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document.write( "* **Decision:** Since the test statistic is not in the rejection region, we **fail to reject the null hypothesis**.\r<BR>\n" );
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document.write( "**Answers**\r<BR>\n" );
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document.write( "* **Two-tailed**<BR>\n" );
document.write( "* **Test statistic:** -1.79<BR>\n" );
document.write( "* **Critical value:** 1.96<BR>\n" );
document.write( "* **Based on this we:** Fail to reject the null hypothesis.<BR>\n" );
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