document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1207795>Question 1207795</A>: <I><SPAN STYLE=\"word-break: break-all;\"> You wish to test the following claim (Ha)at a significance level of a=0.004. \r<BR>\n" );
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document.write( "You believe the population is normally distributed, but you do not know the standard deviation. You obtain the following sample of data:\r<BR>\n" );
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document.write( "data<BR>\n" );
document.write( "54.9<BR>\n" );
document.write( "51.5<BR>\n" );
document.write( "61.5<BR>\n" );
document.write( "66.5<BR>\n" );
document.write( "68.2<BR>\n" );
document.write( "71.5<BR>\n" );
document.write( "76.5<BR>\n" );
document.write( "74.9<BR>\n" );
document.write( "77.9<BR>\n" );
document.write( "91.5<BR>\n" );
document.write( "81.5\r<BR>\n" );
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document.write( "What is the test statistic for this sample? (Report answer accurate to three decimal places.)\r<BR>\n" );
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document.write( "test statistic = <BR>\n" );
document.write( " <BR>\n" );
document.write( "What is the p-value for this sample? (Report answer accurate to four decimal places.)\r<BR>\n" );
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document.write( "p-value = <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 #848039 by </B><A HREF=/tutors/aboutme.mpl?userid=asinus><B>asinus(45)</B></A><img src=\"/images/stars/stars-07.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=asinus><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848039\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Calculate Sample Mean and Standard Deviation**\r<BR>\n" );
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document.write( "* **Sample Mean (x̄):** <BR>\n" );
document.write( "   (54.9 + 51.5 + 61.5 + 66.5 + 68.2 + 71.5 + 76.5 + 74.9 + 77.9 + 91.5 + 81.5) / 11 <BR>\n" );
document.write( "   = 69.27\r<BR>\n" );
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document.write( "* **Sample Standard Deviation (s):** <BR>\n" );
document.write( "   Use a calculator or statistical software to calculate the sample standard deviation. <BR>\n" );
document.write( "   s ≈ 11.783\r<BR>\n" );
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document.write( "**2. Calculate the t-statistic**\r<BR>\n" );
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document.write( "* **Formula:**<BR>\n" );
document.write( "   t = (x̄ - μ₀) / (s / √n) <BR>\n" );
document.write( "   where:<BR>\n" );
document.write( "      * x̄ is the sample mean (69.27)<BR>\n" );
document.write( "      * μ₀ is the hypothesized population mean (you did not provide this value - please specify the value of μ₀ for the accurate calculation)<BR>\n" );
document.write( "      * s is the sample standard deviation (11.783)<BR>\n" );
document.write( "      * n is the sample size (11)\r<BR>\n" );
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document.write( "* **Example:** <BR>\n" );
document.write( "   * Let's assume the hypothesized population mean (μ₀) is 60.<BR>\n" );
document.write( "   * t = (69.27 - 60) / (11.783 / √11) <BR>\n" );
document.write( "   * t ≈ 2.618 \r<BR>\n" );
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document.write( "**3. Calculate the p-value**\r<BR>\n" );
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document.write( "* **Degrees of Freedom:** df = n - 1 = 11 - 1 = 10\r<BR>\n" );
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document.write( "* **Using a t-distribution table or statistical software:** <BR>\n" );
document.write( "   * Find the p-value associated with the calculated t-statistic (2.618) and degrees of freedom (10). <BR>\n" );
document.write( "   * **Note:** Since you did not specify the direction of the alternative hypothesis (Ha), we will assume a two-tailed test.\r<BR>\n" );
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document.write( "* **Example:** <BR>\n" );
document.write( "   * If t = 2.618 and df = 10, the p-value for a two-tailed test is approximately 0.0272.\r<BR>\n" );
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document.write( "**Therefore:**\r<BR>\n" );
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document.write( "* **test statistic = 2.618 (assuming μ₀ = 60)**<BR>\n" );
document.write( "* **p-value = 0.0272 (assuming μ₀ = 60 and a two-tailed test)**\r<BR>\n" );
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document.write( "**Important Notes:**\r<BR>\n" );
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document.write( "* **Hypothesized Mean (μ₀):** You must specify the hypothesized population mean (μ₀) to accurately calculate the t-statistic and p-value. <BR>\n" );
document.write( "* **Software:** Use statistical software (like R, Python, Excel, or SPSS) to perform these calculations more efficiently and accurately.<BR>\n" );
document.write( "* **Interpretation:** If the p-value is less than the significance level (α = 0.004), you would reject the null hypothesis. If the p-value is greater than or equal to α, you would fail to reject the null hypothesis.\r<BR>\n" );
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document.write( "This analysis provides a framework for conducting the t-test. Remember to adjust the calculations based on the specific hypothesized population mean (μ₀) for your analysis.<BR>\n" );
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