document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1177451>Question 1177451</A>: <I><SPAN STYLE=\"word-break: break-all;\"> At the Fredericton High School it is estimated that at most 25% of the students ride bikes to school. Does this seem to be a valid estimate if, in a random sample of 90 high school students, 20 are found to ride bikes to school? Use = 0.01. 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 #850456 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</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=850456\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> We're going to conduct a hypothesis test for a proportion.\r<BR>\n" );
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document.write( "Let $p$ be the true proportion of students who ride bikes to school.\r<BR>\n" );
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document.write( "1. **Hypotheses:**<BR>\n" );
document.write( "   * Null hypothesis: $p \le 0.25$ (The estimate is valid.)<BR>\n" );
document.write( "   * Alternative hypothesis: $p > 0.25$ (The estimate is not valid; the true proportion is higher.)\r<BR>\n" );
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document.write( "2. **Test statistic:**<BR>\n" );
document.write( "   We use the z-statistic for proportions:<BR>\n" );
document.write( "   ```<BR>\n" );
document.write( "   z = \frac{\hat{p} - p_0}{\sqrt{p_0 (1 - p_0) / n}}<BR>\n" );
document.write( "   ```<BR>\n" );
document.write( "   where $\hat{p} = \frac{20}{90} = \frac{2}{9}$ is the sample proportion, $p_0 = 0.25$ is the hypothesized proportion, and $n = 90$ is the sample size.  This gives us<BR>\n" );
document.write( "   ```<BR>\n" );
document.write( "   z = \frac{\frac{2}{9} - 0.25}{\sqrt{0.25 (1 - 0.25) / 90}} \approx -0.23.<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 proportion as extreme as $\hat{p} = \frac{2}{9}$, assuming the null hypothesis is true.  Since this is a right-tailed test, the p-value is<BR>\n" );
document.write( "   ```<BR>\n" );
document.write( "   P(Z > -0.23) = 1 - P(Z \le -0.23) \approx 1 - 0.4090 = 0.5910.<BR>\n" );
document.write( "   ```\r<BR>\n" );
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document.write( "4. **Conclusion:**<BR>\n" );
document.write( "   Since the p-value (0.5910) is greater than $\alpha = 0.01$, we fail to reject the null hypothesis.  There is not enough evidence to conclude that the estimate of at most 25% of students riding bikes to school is not valid. </SPAN>\n" );
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