document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1171395>Question 1171395</A>: <I><SPAN STYLE=\"word-break: break-all;\"> \"\"Untitled,\" by Stephen Chen\r<BR>\n" );
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document.write( "I've often wondered how software is released and sold to the public. Ironically, I work for a company that sells products with known problems. Unfortunately, most of the problems are difficult to create, which makes them difficult to fix. I usually use the test program X, which tests the product, to try to create a specific problem. When the test program is run to make an error occur, the likelihood of generating an error is 1%.\r<BR>\n" );
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document.write( "So, armed with this knowledge, I wrote a new test program Y that will generate the same error that test program X creates, but more often. To find out if my test program is better than the original, so that I can convince the management that I'm right, I ran my test program to find out how often I can generate the same error. When I ran my test program 50 times, I generated the error twice. While this may not seem much better, I think that I can convince the management to use my test program instead of the original test program. Am I right?\r<BR>\n" );
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document.write( "Conduct a hypothesis test at the 5% level.\r<BR>\n" );
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document.write( "Note: If you are using a Student's t-distribution for the problem, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.)\"\r<BR>\n" );
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document.write( "I need help stating the distribution to use for the test. (I must round my answers to four decimal places.)\r<BR>\n" );
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document.write( "format: P'~ N (____, ____) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850948 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=850948\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this hypothesis test and determine the appropriate distribution.\r<BR>\n" );
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document.write( "**1. Define the Problem:**\r<BR>\n" );
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document.write( "* **Null Hypothesis (H0):** The new test program Y is not better than the original program X. The error rate of Y is the same as or less than 1% (0.01).<BR>\n" );
document.write( "* **Alternative Hypothesis (H1):** The new test program Y is better than the original program X. The error rate of Y is greater than 1% (0.01).<BR>\n" );
document.write( "* **Significance Level (α):** 5% or 0.05.\r<BR>\n" );
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document.write( "**2. Data:**\r<BR>\n" );
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document.write( "* **Original Program X:** Error rate (p0) = 0.01<BR>\n" );
document.write( "* **New Program Y:**<BR>\n" );
document.write( "    * Number of trials (n) = 50<BR>\n" );
document.write( "    * Number of errors (x) = 2<BR>\n" );
document.write( "    * Sample proportion (p̂) = x/n = 2/50 = 0.04\r<BR>\n" );
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document.write( "**3. Choose the Appropriate Distribution:**\r<BR>\n" );
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document.write( "* Since we're dealing with proportions and a relatively large sample size (n=50), we can use the **normal distribution** to approximate the binomial distribution.\r<BR>\n" );
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document.write( "**4. Calculate the Standard Error:**\r<BR>\n" );
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document.write( "* The standard error (SE) for a proportion is calculated as:<BR>\n" );
document.write( "    * SE = √(p0(1 - p0) / n)<BR>\n" );
document.write( "    * SE = √(0.01(1 - 0.01) / 50)<BR>\n" );
document.write( "    * SE = √(0.01(0.99) / 50)<BR>\n" );
document.write( "    * SE = √(0.0099 / 50)<BR>\n" );
document.write( "    * SE = √0.000198<BR>\n" );
document.write( "    * SE ≈ 0.01407\r<BR>\n" );
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document.write( "**5. State the Distribution:**\r<BR>\n" );
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document.write( "* P' ~ N (p0, SE^2)<BR>\n" );
document.write( "* P' ~ N (0.01, 0.01407^2)<BR>\n" );
document.write( "* P' ~ N (0.01, 0.000198)\r<BR>\n" );
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document.write( "Therefore, the distribution to use for the test is:\r<BR>\n" );
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document.write( "P' ~ N (0.0100, 0.000198)<BR>\n" );
document.write( " </SPAN>\n" );
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