document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1197674>Question 1197674</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Statistics students believe that the mean score on a first statistics test is 65. The instructor thinks that the mean score is higher. She samples 10 statistics students and obtains the scores:\r<BR>\n" );
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document.write( "Grades	65	64.3	88	69	83.2	68.4	96	64.3	65	62.7\r<BR>\n" );
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document.write( "Test grades are believed to be normally distributed.\r<BR>\n" );
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document.write( "Use a significance level of 5%.<BR>\n" );
document.write( "State the mean of the sample: <BR>\n" );
document.write( "85.08<BR>\n" );
document.write( "Incorrect<BR>\n" );
document.write( "State the standard error of the sample means: <BR>\n" );
document.write( "State the test statistic:  <BR>\n" );
document.write( "t<BR>\n" );
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document.write( "  <BR>\n" );
document.write( "5.935830<BR>\n" );
document.write( "Incorrect<BR>\n" );
document.write( "State the p-value:<BR>\n" );
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document.write( "Answers: <BR>\n" );
document.write( "Mean = <font color=red>72.59</font><BR>\n" );
document.write( "Standard Error = <font color=red>3.769746</font><BR>\n" );
document.write( "Test Statistic = <font color=red>2.01</font><BR>\n" );
document.write( "P-value = <font color=red>0.0377</font><BR>\n" );
document.write( "Conclusion = \"<font color=red>Reject the null</font>\"<BR>\n" );
document.write( "Interpretation = \"<font color=red>The mean score is higher than 65</font>\"\r<BR>\n" );
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document.write( "With exception of the mean, each decimal value is approximate. <BR>\n" );
document.write( "Round however your teacher instructs.\r<BR>\n" );
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document.write( "Explanation:\r<BR>\n" );
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document.write( "Null Hypothesis: mu = 65<BR>\n" );
document.write( "Alternative Hypothesis: mu > 65<BR>\n" );
document.write( "The inequality sign in the alternative hypothesis tells us we have a right-tailed test.\r<BR>\n" );
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document.write( "The data set is<BR>\n" );
document.write( "{65,64.3,88,69,83.2,68.4,96,64.3,65,62.7}<BR>\n" );
document.write( "There are n = 10 items in this data set.\r<BR>\n" );
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document.write( "Add up all of the values in the data set to get <BR>\n" );
document.write( "65+64.3+88+69+83.2+68.4+96+64.3+65+62.7 = 725.9\r<BR>\n" );
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document.write( "Divide that sum by n = 10 to compute the mean<BR>\n" );
document.write( "mean = (sum of values)/n<BR>\n" );
document.write( "mean = (725.9)/10<BR>\n" );
document.write( "<font color=red>mean = 72.59</font><BR>\n" );
document.write( "This value is exact without any rounding done to it.<BR>\n" );
document.write( "This is the sample mean denoted as xbar. Its job is to estimate the population mean mu.\r<BR>\n" );
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document.write( "Use a graphing calculator, spreadsheet, or statistical software to compute the standard deviation of the data set. <BR>\n" );
document.write( "We want the sample standard deviation.<BR>\n" );
document.write( "You should find the sample standard deviation is approximately s = 11.920985\r<BR>\n" );
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document.write( "Let's compute the standard error (SE)<BR>\n" );
document.write( "SE = s/sqrt(n)<BR>\n" );
document.write( "SE = 11.920985/sqrt(10)<BR>\n" );
document.write( "SE = 3.76974645527023<BR>\n" );
document.write( "<font color=red>SE = 3.769746</font><BR>\n" );
document.write( "Round this however your teacher instructs.\r<BR>\n" );
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document.write( "Now we can find the test statistic<BR>\n" );
document.write( "t = (xbar - mu)/SE<BR>\n" );
document.write( "t = (72.59 - 65)/3.769746<BR>\n" );
document.write( "t = 2.01339825017389<BR>\n" );
document.write( "<font color=red>t = 2.01</font><BR>\n" );
document.write( "Many stats textbooks I've come across will have the t statistic accurate to two decimal places. <BR>\n" );
document.write( "But if your teacher requires some other level of precision, then be sure to follow those instructions.\r<BR>\n" );
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document.write( "The sample size is n = 10<BR>\n" );
document.write( "The degrees of freedom (df) is<BR>\n" );
document.write( "df = n-1<BR>\n" );
document.write( "df = 10-1<BR>\n" );
document.write( "df = 9<BR>\n" );
document.write( "Use a stats calculator, spreadsheet, or similar to find the p-value is roughly <font color=red>0.0377</font><BR>\n" );
document.write( "This tells the reader that P(T > 2.01) = 0.0377 approximately when df = 9.<BR>\n" );
document.write( "The p-value may slightly vary depending if you use t = 2.01 or if you use more decimal digits in that test statistic such as t = 2.01339825017389<BR>\n" );
document.write( "Either way, the p-value shouldn't differ too much.\r<BR>\n" );
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document.write( "The p-value being smaller than alpha = 0.05 tells us to <font color=red>reject the null</font> and conclude that mu > 65 appears to be the case. \r<BR>\n" );
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document.write( "Note: we're doing a right tailed test, so be sure to find the area under the T distribution curve to the right of t = 2.01<BR>\n" );
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