document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1178879>Question 1178879</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A manufacturer of children's toys claim that less than 3% of his products are defective.when 500 toys were drawn from a large production rub 5% were found to be defective. Calculate the population interest <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 #850286 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=850286\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this problem and calculate the population proportion of defective toys.\r<BR>\n" );
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document.write( "**1. Define the Variables:**\r<BR>\n" );
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document.write( "* **Claimed Proportion (p₀):** 0.03 (3%) - This is the manufacturer's claim.<BR>\n" );
document.write( "* **Sample Size (n):** 500 toys<BR>\n" );
document.write( "* **Sample Proportion (p̂):** 0.05 (5%) - This is what was found in the sample.\r<BR>\n" );
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document.write( "**2. Understanding the Problem:**\r<BR>\n" );
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document.write( "* The manufacturer claims that p < 0.03.<BR>\n" );
document.write( "* We want to test if the sample proportion (p̂ = 0.05) provides enough evidence to reject the manufacturer's claim.\r<BR>\n" );
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document.write( "**3. Set up the Hypothesis Test:**\r<BR>\n" );
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document.write( "* **Null Hypothesis (H₀):** p ≤ 0.03 (The proportion of defective toys is less than or equal to 3%)<BR>\n" );
document.write( "* **Alternative Hypothesis (H₁):** p > 0.03 (The proportion of defective toys is greater than 3%)<BR>\n" );
document.write( "* This is a right-tailed test.\r<BR>\n" );
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document.write( "**4. Calculate the Test Statistic (z-score):**\r<BR>\n" );
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document.write( "* We'll use the z-test for proportions because we have a large sample size.<BR>\n" );
document.write( "* The formula for the z-score is:\r<BR>\n" );
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document.write( "    z = (p̂ - p₀) / √[p₀(1 - p₀) / n]\r<BR>\n" );
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document.write( "* Plug in the values:\r<BR>\n" );
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document.write( "    z = (0.05 - 0.03) / √[0.03(1 - 0.03) / 500]<BR>\n" );
document.write( "    z = 0.02 / √[0.03(0.97) / 500]<BR>\n" );
document.write( "    z = 0.02 / √[0.0291 / 500]<BR>\n" );
document.write( "    z = 0.02 / √0.0000582<BR>\n" );
document.write( "    z = 0.02 / 0.0076289<BR>\n" );
document.write( "    z ≈ 2.62\r<BR>\n" );
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document.write( "**5. Find the P-value:**\r<BR>\n" );
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document.write( "* We need to find the probability of getting a z-score of 2.62 or higher in a standard normal distribution.<BR>\n" );
document.write( "* Using a z-table or calculator, P(z > 2.62) ≈ 0.0044.\r<BR>\n" );
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document.write( "**6. Conclusion:**\r<BR>\n" );
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document.write( "* The p-value (0.0044) is very small.<BR>\n" );
document.write( "* If we were to use a significance level of 0.05, since 0.0044 < 0.05, we would reject the null hypothesis.<BR>\n" );
document.write( "* This means there is strong evidence to suggest that the true proportion of defective toys is greater than 3%.<BR>\n" );
document.write( "* Therefore, the manufacturer's claim is likely incorrect.<BR>\n" );
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