document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1180121>Question 1180121</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A random sample of 10 observations was drawn from a large normally distributed population. The data is below.\r<BR>\n" );
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document.write( "14	12	20	21	18	22	18	14	12	17 \r<BR>\n" );
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document.write( "Test to determine if we can infer at the 3% significance level that the population mean is not equal to 17, filling in the requested information below.\r<BR>\n" );
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document.write( "A. The value of the standardized test statistic: \r<BR>\n" );
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document.write( "Note: For the next part, your answer should use interval notation. An answer of the form (−∞,𝑎)<BR>\n" );
document.write( " is expressed (-infty, a), an answer of the form (𝑏,∞) is expressed (b, infty), and an answer of the form (−∞,𝑎)∪(𝑏,∞) is expressed (-infty, a)U(b, infty).\r<BR>\n" );
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document.write( "B. The rejection region for the standardized test statistic: \r<BR>\n" );
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document.write( "C. Your decision for the hypothesis test: \r<BR>\n" );
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document.write( "A. Reject 𝐻0<BR>\n" );
document.write( "B. Do Not Reject 𝐻0<BR>\n" );
document.write( "C. Reject 𝐻1<BR>\n" );
document.write( "D. Do Not Reject 𝐻1 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850124 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=850124\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to conduct the hypothesis test:\r<BR>\n" );
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document.write( "**1. State the Hypotheses:**\r<BR>\n" );
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document.write( "* **Null Hypothesis (H0):** The population mean (μ) is equal to 17. (μ = 17)<BR>\n" );
document.write( "* **Alternative Hypothesis (H1):** The population mean (μ) is *not* equal to 17. (μ ≠ 17)  This is a two-tailed test.\r<BR>\n" );
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document.write( "**2. Significance Level:** α = 0.03\r<BR>\n" );
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document.write( "**3. Calculate the Sample Statistics:**\r<BR>\n" );
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document.write( "* **Sample Size (n):** 10<BR>\n" );
document.write( "* **Sample Mean (x̄):**<BR>\n" );
document.write( "    x̄ = (14 + 12 + 20 + 21 + 18 + 22 + 18 + 14 + 12 + 17) / 10 = 168 / 10 = 16.8<BR>\n" );
document.write( "* **Sample Standard Deviation (s):**<BR>\n" );
document.write( "    1. Calculate the squared deviations from the mean:<BR>\n" );
document.write( "    (14-16.8)^2 = 7.84<BR>\n" );
document.write( "    (12-16.8)^2 = 23.04<BR>\n" );
document.write( "    (20-16.8)^2 = 10.24<BR>\n" );
document.write( "    (21-16.8)^2 = 17.64<BR>\n" );
document.write( "    (18-16.8)^2 = 0.04<BR>\n" );
document.write( "    (22-16.8)^2 = 27.04<BR>\n" );
document.write( "    (18-16.8)^2 = 0.04<BR>\n" );
document.write( "    (14-16.8)^2 = 7.84<BR>\n" );
document.write( "    (12-16.8)^2 = 23.04<BR>\n" );
document.write( "    (17-16.8)^2 = 0.04<BR>\n" );
document.write( "    Sum of squared deviations = 116.8\r<BR>\n" );
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document.write( "    s = √[Σ(xᵢ - x̄)² / (n - 1)] = √(116.8 / 9) ≈ √12.98 ≈ 3.603\r<BR>\n" );
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document.write( "**4. Calculate the Test Statistic (t-score):**\r<BR>\n" );
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document.write( "Since the sample size is small (n < 30) and the population standard deviation is unknown, we use a t-test.\r<BR>\n" );
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document.write( "t = (x̄ - μ) / (s / √n)<BR>\n" );
document.write( "t = (16.8 - 17) / (3.603 / √10)<BR>\n" );
document.write( "t = -0.2 / (3.603 / 3.162)<BR>\n" );
document.write( "t = -0.2 / 1.139<BR>\n" );
document.write( "t ≈ -0.176\r<BR>\n" );
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document.write( "**A. The value of the standardized test statistic:** -0.176\r<BR>\n" );
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document.write( "**5. Determine the Degrees of Freedom:**\r<BR>\n" );
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document.write( "Degrees of freedom (df) = n - 1 = 10 - 1 = 9\r<BR>\n" );
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document.write( "**6. Find the Critical t-values:**\r<BR>\n" );
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document.write( "For a two-tailed test with α = 0.03 and df = 9, we need to find the t-values that correspond to α/2 = 0.015 in each tail. Using a t-table or calculator, we find the critical t-values are approximately ±2.821.\r<BR>\n" );
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document.write( "**B. The rejection region for the standardized test statistic:** (-infty, -2.821)U(2.821, infty)\r<BR>\n" );
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document.write( "**7. Make a Decision:**\r<BR>\n" );
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document.write( "Compare the calculated t-statistic (-0.176) to the critical t-values (±2.821).\r<BR>\n" );
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document.write( "* -2.821 < -0.176 < 2.821\r<BR>\n" );
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document.write( "Since the calculated t-statistic falls *within* the range of the critical t-values, we *fail to reject* the null hypothesis.\r<BR>\n" );
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document.write( "**C. Your decision for the hypothesis test:** B. Do Not Reject H0<BR>\n" );
document.write( " </SPAN>\n" );
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