document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1176438>Question 1176438</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 3.	A principal states that in his school, the average number of days per year missed by a student due to illness is less than 10. The ff. data show the number of days missed by 40 students a year. Is there sufficient evidence to believe the principal’s claim at alpha=.05? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850578 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=850578\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To test the principal's claim, we'll perform a one-tailed t-test. Since you haven't provided the actual data, I'll outline the steps and provide a general solution. You'll need to plug in the data to complete the calculations.\r<BR>\n" );
document.write( "\n" );
document.write( "**1. State the Hypotheses:**\r<BR>\n" );
document.write( "\n" );
document.write( "* **Null Hypothesis (H0):** μ ≥ 10 (The average number of days missed is greater than or equal to 10)<BR>\n" );
document.write( "* **Alternative Hypothesis (H1):** μ < 10 (The average number of days missed is less than 10)\r<BR>\n" );
document.write( "\n" );
document.write( "**2. Determine the Significance Level:**\r<BR>\n" );
document.write( "\n" );
document.write( "* α = 0.05\r<BR>\n" );
document.write( "\n" );
document.write( "**3. Calculate the Sample Statistics:**\r<BR>\n" );
document.write( "\n" );
document.write( "* **Sample Size (n):** 40<BR>\n" );
document.write( "* **Calculate the Sample Mean (x̄):** Sum all the data points and divide by 40.<BR>\n" );
document.write( "* **Calculate the Sample Standard Deviation (s):** Use the formula:<BR>\n" );
document.write( "    * s = √[Σ(xᵢ - x̄)² / (n - 1)]\r<BR>\n" );
document.write( "\n" );
document.write( "**4. Calculate the Test Statistic (t):**\r<BR>\n" );
document.write( "\n" );
document.write( "* t = (x̄ - μ) / (s / √n)<BR>\n" );
document.write( "    * Where:<BR>\n" );
document.write( "        * x̄ = sample mean<BR>\n" );
document.write( "        * μ = population mean (10)<BR>\n" );
document.write( "        * s = sample standard deviation<BR>\n" );
document.write( "        * n = sample size\r<BR>\n" );
document.write( "\n" );
document.write( "**5. Determine the Critical Value:**\r<BR>\n" );
document.write( "\n" );
document.write( "* Since we are performing a left-tailed test at α = 0.05 with df = n - 1 = 39, we need to find the critical t-value from the t-distribution table.<BR>\n" );
document.write( "* For df = 39 and α = 0.05 (one-tailed), the critical t-value is approximately -1.684.\r<BR>\n" );
document.write( "\n" );
document.write( "**6. Make a Decision:**\r<BR>\n" );
document.write( "\n" );
document.write( "* **If the calculated t-statistic is less than the critical t-value (-1.684), reject the null hypothesis.** This means there is sufficient evidence to support the principal's claim.<BR>\n" );
document.write( "* **If the calculated t-statistic is greater than or equal to the critical t-value, fail to reject the null hypothesis.** This means there is not sufficient evidence to support the principal's claim.\r<BR>\n" );
document.write( "\n" );
document.write( "**Example (Illustrative - You need to plug in your data):**\r<BR>\n" );
document.write( "\n" );
document.write( "Let's say, after calculating from your data, you find:\r<BR>\n" );
document.write( "\n" );
document.write( "* x̄ = 8.5<BR>\n" );
document.write( "* s = 4.0\r<BR>\n" );
document.write( "\n" );
document.write( "Then:\r<BR>\n" );
document.write( "\n" );
document.write( "* t = (8.5 - 10) / (4.0 / √40)<BR>\n" );
document.write( "* t = -1.5 / (4.0 / 6.3245)<BR>\n" );
document.write( "* t = -1.5 / 0.63245<BR>\n" );
document.write( "* t = -2.372\r<BR>\n" );
document.write( "\n" );
document.write( "In this example:\r<BR>\n" );
document.write( "\n" );
document.write( "* -2.372 < -1.684\r<BR>\n" );
document.write( "\n" );
document.write( "Therefore, you would reject the null hypothesis.\r<BR>\n" );
document.write( "\n" );
document.write( "**Conclusion:**\r<BR>\n" );
document.write( "\n" );
document.write( "* **If your calculated t-statistic is less than -1.684, there is sufficient evidence to support the principal's claim that the average number of days missed is less than 10 at α = 0.05.**<BR>\n" );
document.write( "* **If your calculated t-statistic is greater than or equal to -1.684, there is not sufficient evidence to support the principal's claim at α = 0.05.** </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );