document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1167903>Question 1167903</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 2.	A Corkill Machine is set ti fill a small bottle with 9.0 grams of medicine.  It is claimed that the mean weight is less 9.0 grams. The hypothesis is to be tested at the 0.01 level.  A sample revealed these weights (in grams): 9.2,  8.7,  8.9,  8.6,  8.8,  8.5,  8.7, and 9.0.<BR>\n" );
document.write( "a.	State the null and alternate hypotheses.<BR>\n" );
document.write( "b.	How many degrees of freedom are there?<BR>\n" );
document.write( "c.	Give the decision rule.<BR>\n" );
document.write( "d.	Compute t-value and arrive at a decision.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852081 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=852081\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This problem requires a **one-sample t-test** because we are comparing a sample mean to a known population mean (or hypothesized mean) when the population standard deviation is unknown and the sample size is small ($n < 30$).\r<BR>\n" );
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document.write( "**Given Data:**<BR>\n" );
document.write( "* Hypothesized population mean ($\mu_0$): 9.0 grams<BR>\n" );
document.write( "* Sample weights: $X = \{9.2, 8.7, 8.9, 8.6, 8.8, 8.5, 8.7, 9.0\}$<BR>\n" );
document.write( "* Sample size ($n$): 8<BR>\n" );
document.write( "* Significance level ($\alpha$): 0.01\r<BR>\n" );
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document.write( "**a. State the null and alternate hypotheses.**\r<BR>\n" );
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document.write( "* **Null Hypothesis ($H_0$):** The mean weight of medicine filled by the Corkill Machine is equal to or greater than 9.0 grams.<BR>\n" );
document.write( "    $H_0: \mu \ge 9.0$<BR>\n" );
document.write( "* **Alternate Hypothesis ($H_1$):** The mean weight of medicine filled by the Corkill Machine is less than 9.0 grams.<BR>\n" );
document.write( "    $H_1: \mu < 9.0$\r<BR>\n" );
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document.write( "This is a **one-tailed (left-tailed) test**.\r<BR>\n" );
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document.write( "**b. How many degrees of freedom are there?**\r<BR>\n" );
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document.write( "For a one-sample t-test, the degrees of freedom ($df$) are calculated as $n - 1$, where $n$ is the sample size.\r<BR>\n" );
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document.write( "* $df = 8 - 1 = 7$\r<BR>\n" );
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document.write( "There are **7 degrees of freedom**.\r<BR>\n" );
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document.write( "**c. Give the decision rule.**\r<BR>\n" );
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document.write( "The decision rule is based on comparing the calculated t-value to a critical t-value from the t-distribution table.\r<BR>\n" );
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document.write( "* **Significance Level ($\alpha$):** 0.01<BR>\n" );
document.write( "* **Degrees of Freedom ($df$):** 7<BR>\n" );
document.write( "* **Type of Test:** One-tailed (left-tailed)\r<BR>\n" );
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document.write( "From the t-distribution table, the critical t-value for $\alpha = 0.01$ and $df = 7$ for a one-tailed test is approximately **-2.998**.\r<BR>\n" );
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document.write( "**Decision Rule:** Reject the null hypothesis ($H_0$) if the calculated t-value is less than -2.998. Otherwise, fail to reject $H_0$.\r<BR>\n" );
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document.write( "**d. Compute t-value and arrive at a decision.**\r<BR>\n" );
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document.write( "**1. Calculate Sample Statistics:**\r<BR>\n" );
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document.write( "* **Sample Mean ($\bar{x}$):**<BR>\n" );
document.write( "    $\sum X = 9.2 + 8.7 + 8.9 + 8.6 + 8.8 + 8.5 + 8.7 + 9.0 = 70.4$<BR>\n" );
document.write( "    $\bar{x} = \frac{\sum X}{n} = \frac{70.4}{8} = 8.8$\r<BR>\n" );
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document.write( "* **Sample Standard Deviation ($s$):**<BR>\n" );
document.write( "    First, calculate $\sum (X - \bar{x})^2$:<BR>\n" );
document.write( "    * $(9.2 - 8.8)^2 = (0.4)^2 = 0.16$<BR>\n" );
document.write( "    * $(8.7 - 8.8)^2 = (-0.1)^2 = 0.01$<BR>\n" );
document.write( "    * $(8.9 - 8.8)^2 = (0.1)^2 = 0.01$<BR>\n" );
document.write( "    * $(8.6 - 8.8)^2 = (-0.2)^2 = 0.04$<BR>\n" );
document.write( "    * $(8.8 - 8.8)^2 = (0.0)^2 = 0.00$<BR>\n" );
document.write( "    * $(8.5 - 8.8)^2 = (-0.3)^2 = 0.09$<BR>\n" );
document.write( "    * $(8.7 - 8.8)^2 = (-0.1)^2 = 0.01$<BR>\n" );
document.write( "    * $(9.0 - 8.8)^2 = (0.2)^2 = 0.04$<BR>\n" );
document.write( "    $\sum (X - \bar{x})^2 = 0.16 + 0.01 + 0.01 + 0.04 + 0.00 + 0.09 + 0.01 + 0.04 = 0.36$\r<BR>\n" );
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document.write( "    $s = \sqrt{\frac{\sum (X - \bar{x})^2}{n-1}} = \sqrt{\frac{0.36}{8-1}} = \sqrt{\frac{0.36}{7}} \approx \sqrt{0.051428} \approx 0.22678$\r<BR>\n" );
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document.write( "**2. Compute the t-value:**\r<BR>\n" );
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document.write( "The formula for the t-statistic is:<BR>\n" );
document.write( "$t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$<BR>\n" );
document.write( "$t = \frac{8.8 - 9.0}{0.22678 / \sqrt{8}}$<BR>\n" );
document.write( "$t = \frac{-0.2}{0.22678 / 2.8284}$<BR>\n" );
document.write( "$t = \frac{-0.2}{0.08018}$<BR>\n" );
document.write( "$t \approx -2.494$\r<BR>\n" );
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document.write( "**3. Arrive at a Decision:**\r<BR>\n" );
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document.write( "* Calculated t-value: $-2.494$<BR>\n" );
document.write( "* Critical t-value: $-2.998$\r<BR>\n" );
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document.write( "Since the calculated t-value ($-2.494$) is **greater than** the critical t-value ($-2.998$), it does not fall into the rejection region.\r<BR>\n" );
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document.write( "**Decision:** Fail to reject the null hypothesis ($H_0$).\r<BR>\n" );
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document.write( "**Conclusion:**<BR>\n" );
document.write( "At the 0.01 significance level, there is **not enough statistical evidence** to support the claim that the mean weight of medicine filled by the Corkill Machine is less than 9.0 grams. </SPAN>\n" );
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