document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1168567>Question 1168567</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A teacher expected students to take about 20 mins on their online exam. She noticed that a lot of her students were taking less time and she is wondering if there is a systematic difference or if what she is seeing is just due to chance. Did her class of 48 students differ significantly from her typical average of 20 mins?<BR>\n" );
document.write( "Question: <BR>\n" );
document.write( "1. Statistical Hypotheses?<BR>\n" );
document.write( "2. reject the null hypothesis at the .05 level of significance if t ≥ __<BR>\n" );
document.write( " or t ≤ __ given __ df.<BR>\n" );
document.write( "3. what is your decision regarding whether she should reject or accept the null hypothesis.<BR>\n" );
document.write( "4. How do you interpret your decision?<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851559 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2189)</B></A><img src=\"/images/stars/stars-13.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=851559\">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" );
document.write( "\n" );
document.write( "**1. Statistical Hypotheses**\r<BR>\n" );
document.write( "\n" );
document.write( "* **Null Hypothesis (H₀):** The mean time taken by the students is 20 minutes (μ = 20).<BR>\n" );
document.write( "* **Alternative Hypothesis (H₁):** The mean time taken by the students is different from 20 minutes (μ ≠ 20).\r<BR>\n" );
document.write( "\n" );
document.write( "This is a two-tailed test because we are testing if the time is *different* from 20 minutes, not specifically less than or greater than.\r<BR>\n" );
document.write( "\n" );
document.write( "**2. Reject the Null Hypothesis**\r<BR>\n" );
document.write( "\n" );
document.write( "* Significance level (α) = 0.05<BR>\n" );
document.write( "* Sample size (n) = 48<BR>\n" );
document.write( "* Degrees of freedom (df) = n - 1 = 48 - 1 = 47\r<BR>\n" );
document.write( "\n" );
document.write( "Since the sample size is large (n > 30), we can use the t-distribution or approximate it with the z-distribution. However, it's safer to use the t-distribution with df = 47.\r<BR>\n" );
document.write( "\n" );
document.write( "* Using a t-table or calculator:<BR>\n" );
document.write( "    * t(0.025, 47) ≈ ±2.012 (This is the critical t-value for a two-tailed test with α = 0.05 and df = 47).\r<BR>\n" );
document.write( "\n" );
document.write( "* Reject H₀ if t ≥ 2.012 or t ≤ -2.012.\r<BR>\n" );
document.write( "\n" );
document.write( "**3. Decision (We need More Information to Make a Decision)**\r<BR>\n" );
document.write( "\n" );
document.write( "To make a decision, we need the following information:\r<BR>\n" );
document.write( "\n" );
document.write( "* **Sample mean (x̄):** The average time her class actually took.<BR>\n" );
document.write( "* **Sample standard deviation (s):** The standard deviation of the times her class took.\r<BR>\n" );
document.write( "\n" );
document.write( "Without this information, we cannot calculate the t-statistic and make a decision.\r<BR>\n" );
document.write( "\n" );
document.write( "**Example Scenario (to illustrate the decision-making process):**\r<BR>\n" );
document.write( "\n" );
document.write( "Let's assume:\r<BR>\n" );
document.write( "\n" );
document.write( "* Sample mean (x̄) = 18 minutes<BR>\n" );
document.write( "* Sample standard deviation (s) = 5 minutes\r<BR>\n" );
document.write( "\n" );
document.write( "Now, we can calculate the t-statistic:\r<BR>\n" );
document.write( "\n" );
document.write( "* t = (x̄ - μ) / (s / √n)<BR>\n" );
document.write( "* t = (18 - 20) / (5 / √48)<BR>\n" );
document.write( "* t = -2 / (5 / 6.928)<BR>\n" );
document.write( "* t = -2 / 0.7216<BR>\n" );
document.write( "* t ≈ -2.772\r<BR>\n" );
document.write( "\n" );
document.write( "In this example, t = -2.772. Since -2.772 ≤ -2.012, we would reject the null hypothesis.\r<BR>\n" );
document.write( "\n" );
document.write( "**4. Interpretation (Based on the Example Scenario)**\r<BR>\n" );
document.write( "\n" );
document.write( "If we used the example data, the interpretation would be:\r<BR>\n" );
document.write( "\n" );
document.write( "* The results of the t-test indicate that there is a statistically significant difference in the average time taken by the students compared to the teacher's expected 20 minutes.<BR>\n" );
document.write( "* Specifically, the students took significantly less time than expected (18 minutes on average in our example).<BR>\n" );
document.write( "* This suggests that the teacher's observation of students taking less time is not likely due to chance and that there is a systematic difference.\r<BR>\n" );
document.write( "\n" );
document.write( "**Important Note:**\r<BR>\n" );
document.write( "\n" );
document.write( "* The actual decision and interpretation depend on the sample mean and standard deviation, which were not provided in the original question.<BR>\n" );
document.write( "* If you provide the sample mean and standard deviation, I can give you a more specific answer.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );