document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1181618>Question 1181618</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Modern oil tankers weigh more than a 1/2 million tons and have lengths up to one - fourth mile. Such massive ships require a distance of 5.0 km about (3.0 mi) and a time of 20 minutes to come to a stop from top speed of 30 km/h.what is the magnitude of such ship's average acceleration in m/s squared in coming to a stop?<BR>\n" );
document.write( "B.  What is the magnitude of the ship's average velocity in m/s </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849991 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=849991\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's a complete hypothesis test for the manufacturer's claim:\r<BR>\n" );
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document.write( "**a) Solution:**\r<BR>\n" );
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document.write( "We will use a two-sample z-test for the difference between means since the sample sizes are large (n > 30) and the population standard deviations are known.\r<BR>\n" );
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document.write( "**b) Parameter under study:**\r<BR>\n" );
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document.write( "The parameter under study is the difference between the average tensile strength of yarn A (μA) and the average tensile strength of yarn B (μB).  We are interested in whether μA - μB < 12.\r<BR>\n" );
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document.write( "**c) Statement of Hypotheses:**\r<BR>\n" );
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document.write( "*   Null hypothesis (H₀): μA - μB ≥ 12  (The difference is greater than or equal to 12 kg)<BR>\n" );
document.write( "*   Alternative hypothesis (H₁ or Ha): μA - μB < 12 (The manufacturer's claim: The difference is less than 12 kg)\r<BR>\n" );
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document.write( "**d) Significance level:**\r<BR>\n" );
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document.write( "α = 0.005\r<BR>\n" );
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document.write( "**e) Test statistic:**\r<BR>\n" );
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document.write( "The test statistic for a two-sample z-test is:\r<BR>\n" );
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document.write( "z = [(xA - xB) - (μA - μB)₀] / sqrt(σA²/nA + σB²/nB)\r<BR>\n" );
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document.write( "Where:<BR>\n" );
document.write( "* xA and xB are the sample means<BR>\n" );
document.write( "* (μA - μB)₀ is the hypothesized difference under H₀ (12)<BR>\n" );
document.write( "* σA and σB are the population standard deviations<BR>\n" );
document.write( "* nA and nB are the sample sizes\r<BR>\n" );
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document.write( "z = [(86.7 - 77.8) - 12] / sqrt((6.28²/50) + (5.61²/45))<BR>\n" );
document.write( "z = [-3.1] / sqrt(0.788 + 0.700)<BR>\n" );
document.write( "z = -3.1 / sqrt(1.488)<BR>\n" );
document.write( "z ≈ -3.1 / 1.22<BR>\n" );
document.write( "z ≈ -2.54\r<BR>\n" );
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document.write( "**f) Rejection region:**\r<BR>\n" );
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document.write( "Since this is a one-tailed test (less than), we look for the critical z-value that corresponds to α = 0.005 in the left tail of the standard normal distribution. Using a z-table or calculator, we find that the critical z-value is approximately -2.576.\r<BR>\n" );
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document.write( "The rejection region is z < -2.576.\r<BR>\n" );
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document.write( "**g) Decision making:**\r<BR>\n" );
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document.write( "Our calculated test statistic (z ≈ -2.54) falls *within* the rejection region (z < -2.576) .\r<BR>\n" );
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document.write( "Therefore, we *reject* the null hypothesis.\r<BR>\n" );
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document.write( "**h) Conclusion:**\r<BR>\n" );
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document.write( "There is sufficient evidence at the 0.005 significance level to support the manufacturer's claim that the average tensile strength of yarn A is less than the average tensile strength of yarn B by less than 12 kilograms.<BR>\n" );
document.write( " </SPAN>\n" );
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