document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1162408>Question 1162408</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The mean length of a small counterbalance bar is 43 millimeters. The production supervisor<BR>\n" );
document.write( "is concerned that the adjustments of the machine producing the bar have changed. He asks<BR>\n" );
document.write( "the engineering to investigate. Engineering selects a random sample of 30 bars and measure<BR>\n" );
document.write( "each. The results are reported below in millimeters.<BR>\n" );
document.write( "42 39 42 45 43 40 39 41 40 42<BR>\n" );
document.write( "43 42 45 47 38 46 48 42 47 41<BR>\n" );
document.write( "40 44 43 42 40 41 42 39 40 41<BR>\n" );
document.write( "Is it reasonable to conclude that the mean length of the bars reduced? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #854405 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2264)</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=854405\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> ```python?code_reference&code_event_index=3<BR>\n" );
document.write( "import numpy as np<BR>\n" );
document.write( "from scipy import stats\r<BR>\n" );
document.write( "\n" );
document.write( "data = [<BR>\n" );
document.write( "    42, 39, 42, 45, 43, 40, 39, 41, 40, 42,<BR>\n" );
document.write( "    43, 42, 45, 47, 38, 46, 48, 42, 47, 41,<BR>\n" );
document.write( "    40, 44, 43, 42, 40, 41, 42, 39, 40, 41<BR>\n" );
document.write( "]\r<BR>\n" );
document.write( "\n" );
document.write( "mu_0 = 43<BR>\n" );
document.write( "n = len(data)<BR>\n" );
document.write( "x_bar = np.mean(data)<BR>\n" );
document.write( "s = np.std(data, ddof=1)<BR>\n" );
document.write( "se = s / np.sqrt(n)<BR>\n" );
document.write( "t_stat = (x_bar - mu_0) / se<BR>\n" );
document.write( "df = n - 1\r<BR>\n" );
document.write( "\n" );
document.write( "# One-tailed test (Reduced mean: HA: mu < 43)<BR>\n" );
document.write( "p_value = stats.t.cdf(t_stat, df)\r<BR>\n" );
document.write( "\n" );
document.write( "print(f\"{x_bar=}\")<BR>\n" );
document.write( "print(f\"{s=}\")<BR>\n" );
document.write( "print(f\"{t_stat=}\")<BR>\n" );
document.write( "print(f\"{p_value=}\")\r<BR>\n" );
document.write( "\n" );
document.write( "```<BR>\n" );
document.write( "```text?code_stdout&code_event_index=3<BR>\n" );
document.write( "x_bar=42.13333333333333<BR>\n" );
document.write( "s=2.582879085016581<BR>\n" );
document.write( "t_stat=-1.8378440010020713<BR>\n" );
document.write( "p_value=0.038175278168928364\r<BR>\n" );
document.write( "\n" );
document.write( "```<BR>\n" );
document.write( "To determine if the mean length of the bars has reduced, we conduct a **one-sample t-test**.\r<BR>\n" );
document.write( "\n" );
document.write( "### 1. State the Hypotheses<BR>\n" );
document.write( "* **Null Hypothesis ($H_0$):** $\mu \geq 43$ (The mean length has not reduced)<BR>\n" );
document.write( "* **Alternative Hypothesis ($H_A$):** $\mu < 43$ (The mean length has reduced)\r<BR>\n" );
document.write( "\n" );
document.write( "### 2. Identify Sample Statistics<BR>\n" );
document.write( "From the provided data of $n = 30$ bars:<BR>\n" );
document.write( "* **Sample Mean ($\bar{x}$):** $42.13$ mm<BR>\n" );
document.write( "* **Sample Standard Deviation ($s$):** $2.58$ mm<BR>\n" );
document.write( "* **Hypothesized Mean ($\mu_0$):** $43$ mm\r<BR>\n" );
document.write( "\n" );
document.write( "### 3. Calculate the Test Statistic ($t$)<BR>\n" );
document.write( "The $t$-statistic measures how many standard errors the sample mean is from the hypothesized mean:<BR>\n" );
document.write( "$$t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} = \frac{42.13 - 43}{2.58 / \sqrt{30}} \approx -1.838$$\r<BR>\n" );
document.write( "\n" );
document.write( "### 4. Determine the p-value<BR>\n" );
document.write( "Using the $t$-distribution with $df = n - 1 = 29$:<BR>\n" );
document.write( "* **$p$-value:** $\approx 0.0382$\r<BR>\n" );
document.write( "\n" );
document.write( "### 5. Conclusion<BR>\n" );
document.write( "At a standard significance level of **$\alpha = 0.05$**:<BR>\n" );
document.write( "* Since the $p$-value ($0.0382$) is **less than** $0.05$, we **reject the null hypothesis**.\r<BR>\n" );
document.write( "\n" );
document.write( "**Final Result:** Yes, it is **reasonable to conclude** that the mean length of the bars has reduced. The statistical evidence indicates that the decrease from $43$ mm to $42.13$ mm is significant enough that it is unlikely to have occurred by random chance alone. \r<BR>\n" );
document.write( "\n" );
document.write( "> **Note:** If a more stringent significance level like $\alpha = 0.01$ were used, we would fail to reject the null hypothesis, as $0.0382 > 0.01$. However, in most industrial engineering contexts, $0.05$ is the standard threshold for investigation. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );