document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1177447>Question 1177447</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The Hinton Valley Bottling Company distributes cola in bottles labeled 32oz. An inspector randomly chooses 50 of these bottles, measures their contents, and obtains a sample mean of 31.4oz. and a sample standard deviation of 1.75oz. Using a 0.01 level of significance, test the inspector’s claim that the company is cheating the consumers. Find the p-value for this test. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850459 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=850459\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> We're going to conduct a hypothesis test for a mean.\r<BR>\n" );
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document.write( "Let $\mu$ be the true mean volume of cola in the bottles.\r<BR>\n" );
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document.write( "1. **Hypotheses:**<BR>\n" );
document.write( "   * Null hypothesis: $\mu = 32$ (The company is not cheating.)<BR>\n" );
document.write( "   * Alternative hypothesis: $\mu < 32$ (The company is cheating.)\r<BR>\n" );
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document.write( "2. **Test statistic:**<BR>\n" );
document.write( "   We use the t-statistic, since the population standard deviation is unknown:<BR>\n" );
document.write( "   ```<BR>\n" );
document.write( "   t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}<BR>\n" );
document.write( "   ```<BR>\n" );
document.write( "   where $\bar{x} = 31.4$ is the sample mean, $\mu_0 = 32$ is the hypothesized mean, $s = 1.75$ is the sample standard deviation, and $n = 50$ is the sample size.  This gives us<BR>\n" );
document.write( "   ```<BR>\n" );
document.write( "   t = \frac{31.4 - 32}{1.75 / \sqrt{50}} \approx -2.42.<BR>\n" );
document.write( "   ```\r<BR>\n" );
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document.write( "3. **P-value:**<BR>\n" );
document.write( "   The p-value is the probability of observing a sample mean as extreme as $\bar{x} = 31.4$, assuming the null hypothesis is true.  Since this is a left-tailed test, the p-value is<BR>\n" );
document.write( "   ```<BR>\n" );
document.write( "   P(T < -2.42) \approx 0.0096,<BR>\n" );
document.write( "   ```<BR>\n" );
document.write( "   where $T$ has a t-distribution with $n - 1 = 49$ degrees of freedom.\r<BR>\n" );
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document.write( "4. **Conclusion:**<BR>\n" );
document.write( "   Since the p-value (0.0096) is less than $\alpha = 0.01$, we reject the null hypothesis.  There is enough evidence to conclude that the company is cheating the consumers. </SPAN>\n" );
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