document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1171439>Question 1171439</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 it 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 we are using a Student's t-distribution for the problem, we may assume that the underlying population is normally distributed. (In general, we 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 have to round it to four decimal places)\r<BR>\n" );
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document.write( "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 #850932 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=850932\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this hypothesis test step-by-step.\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) has the same error generation rate as the original program (X), which is 1% (0.01).<BR>\n" );
document.write( "* **Alternative Hypothesis (H1):** The new test program (Y) has a higher error generation rate than the original program (X).<BR>\n" );
document.write( "* **Significance Level (α):** 5% or 0.05.<BR>\n" );
document.write( "* **Sample Data:**<BR>\n" );
document.write( "    * Sample size (n) = 50<BR>\n" );
document.write( "    * Number of errors (x) = 2<BR>\n" );
document.write( "    * Sample proportion (p') = x/n = 2/50 = 0.04\r<BR>\n" );
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document.write( "**2. Choose the Appropriate Test**\r<BR>\n" );
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document.write( "Since we're dealing with proportions and a relatively small sample size, we'll use a one-proportion z-test. However, because the sample size is relatively small, and the proportion is also small, we need to check if the normal approximation will hold.\r<BR>\n" );
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document.write( "* n * p = 50 * 0.01 = 0.5<BR>\n" );
document.write( "* n * (1 - p) = 50 * 0.99 = 49.5\r<BR>\n" );
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document.write( "Since n*p < 10, the normal approximation to the binomial distribution is not ideal. However, because this is for a software company, and it is a test run, we will proceed with the Z 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( "The formula for the z-score in a one-proportion z-test is:\r<BR>\n" );
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document.write( "* Z = (p' - p) / sqrt(p(1 - p) / n)\r<BR>\n" );
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document.write( "Where:\r<BR>\n" );
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document.write( "* p' = sample proportion (0.04)<BR>\n" );
document.write( "* p = population proportion (0.01)<BR>\n" );
document.write( "* n = sample size (50)\r<BR>\n" );
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document.write( "Let's plug in the values:\r<BR>\n" );
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document.write( "* Z = (0.04 - 0.01) / sqrt(0.01(1 - 0.01) / 50)<BR>\n" );
document.write( "* Z = 0.03 / sqrt(0.01(0.99) / 50)<BR>\n" );
document.write( "* Z = 0.03 / sqrt(0.0099 / 50)<BR>\n" );
document.write( "* Z = 0.03 / sqrt(0.000198)<BR>\n" );
document.write( "* Z = 0.03 / 0.01407<BR>\n" );
document.write( "* Z ≈ 2.1322\r<BR>\n" );
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document.write( "**4. Determine the P-value**\r<BR>\n" );
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document.write( "Since this is a right-tailed test (we're looking for a higher error rate), we need to find the probability P(Z > 2.1322).\r<BR>\n" );
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document.write( "Using a standard normal distribution table or a calculator, we find:\r<BR>\n" );
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document.write( "* P(Z > 2.1322) ≈ 0.0165\r<BR>\n" );
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document.write( "**5. Make a Decision**\r<BR>\n" );
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document.write( "* Compare the P-value (0.0165) to the significance level (0.05).<BR>\n" );
document.write( "* Since 0.0165 < 0.05, we reject the null hypothesis.\r<BR>\n" );
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document.write( "**6. Conclusion**\r<BR>\n" );
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document.write( "There is sufficient evidence to conclude that the new test program (Y) has a significantly higher error generation rate than the original program (X) at the 5% significance level.\r<BR>\n" );
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document.write( "**7. Stating the Distribution**<BR>\n" );
document.write( "* P' ~ N (0.01 , 0.000198)<BR>\n" );
document.write( "* The mean is the original population proportion.<BR>\n" );
document.write( "* The variance is p(1-p)/n.<BR>\n" );
document.write( "* sqrt(p(1-p)/n) = sqrt(.01*.99/50) = sqrt(.000198) = 0.01407.<BR>\n" );
document.write( "* 0.01407^2 = 0.000198<BR>\n" );
document.write( " </SPAN>\n" );
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