document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1181114>Question 1181114</A>: <I><SPAN STYLE=\"word-break: break-all;\"> (a) An advertising company wants to estimate with 98% confidence interval, the number of times a website is hit during an hour. It is determined that = 26. How large a sample should the company take, if it wishes that the margin of error should not exceed 10? [04]<BR>\n" );
document.write( "(b) If we take a sample from an infinite population, what will happen to the standard error of the mean when the sample size is increased from 60 to 240? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850073 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=850073\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **(a) Sample size calculation:**\r<BR>\n" );
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document.write( "1. **Identify the given information:**\r<BR>\n" );
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document.write( "*   Confidence level = 98%<BR>\n" );
document.write( "*   Margin of error (E) = 10<BR>\n" );
document.write( "*   Population standard deviation (σ) = 26\r<BR>\n" );
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document.write( "2. **Find the critical z-value:**\r<BR>\n" );
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document.write( "For a 98% confidence level, α = 1 - 0.98 = 0.02.  α/2 = 0.01.  The critical z-value (z*) corresponding to 0.01 in the tail of the standard normal distribution is approximately 2.33.\r<BR>\n" );
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document.write( "3. **Use the sample size formula:**\r<BR>\n" );
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document.write( "n = (z* * σ / E)²\r<BR>\n" );
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document.write( "4. **Plug in the values:**\r<BR>\n" );
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document.write( "n = (2.33 * 26 / 10)²<BR>\n" );
document.write( "n = (60.58 / 10)²<BR>\n" );
document.write( "n = 6.058²<BR>\n" );
document.write( "n ≈ 36.7\r<BR>\n" );
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document.write( "5. **Round up:** Since the sample size must be a whole number, always round up to the nearest integer. Therefore, the company should take a sample of at least 37.\r<BR>\n" );
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document.write( "**(b) Effect of sample size on standard error:**\r<BR>\n" );
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document.write( "The standard error of the mean (SEM) is calculated as:\r<BR>\n" );
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document.write( "SEM = σ / √n\r<BR>\n" );
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document.write( "Where:\r<BR>\n" );
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document.write( "*   σ = population standard deviation<BR>\n" );
document.write( "*   n = sample size\r<BR>\n" );
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document.write( "If the sample size is increased from n₁ = 60 to n₂ = 240, let's see what happens to the SEM.\r<BR>\n" );
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document.write( "*   Initial SEM (SEM₁) = σ / √60<BR>\n" );
document.write( "*   New SEM (SEM₂) = σ / √240\r<BR>\n" );
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document.write( "We can rewrite √240 as √(4 * 60) = 2√60\r<BR>\n" );
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document.write( "So, SEM₂ = σ / (2√60) = (1/2) * (σ / √60) = (1/2) * SEM₁\r<BR>\n" );
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document.write( "Therefore, when the sample size is increased from 60 to 240 (a four-fold increase), the standard error of the mean is *halved*.<BR>\n" );
document.write( " </SPAN>\n" );
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