document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1197672>Question 1197672</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The average house has 15 paintings on its walls.  Is the mean different for houses owned by teachers? The data show the results of a survey of 15 teachers who were asked how many paintings they have in their houses. Assume that the distribution of the population is normal.\r<BR>\n" );
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document.write( "15, 13, 15, 15, 14, 12, 12, 14, 14, 15, 14, 15, 16, 13, 13\r<BR>\n" );
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document.write( "What can be concluded at the   <BR>\n" );
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document.write( "  = 0.05 level of significance? </SPAN>\n" );
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document.write( "Answer: The number of paintings that teachers have is different from 15 paintings.\r<BR>\n" );
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document.write( "Explanation:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=mu ALIGN=MIDDLE  ALT=\"mu\"   DISABLED_style= \"max-width:90%; height:auto;\" > = Greek letter mu, represents the population mean<BR>\n" );
document.write( "More specifically, mu is the population mean of the number of paintings per teacher household.\r<BR>\n" );
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document.write( "Null Hypothesis: mu = 15<BR>\n" );
document.write( "Alternative Hypothesis: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=mu+%3C%3E+15 ALIGN=MIDDLE  ALT=\"mu+%3C%3E+15\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The \"not equal\" sign in the alternative says we have a two-tailed test.\r<BR>\n" );
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document.write( "The data set is<BR>\n" );
document.write( "{15, 13, 15, 15, 14, 12, 12, 14, 14, 15, 14, 15, 16, 13, 13}<BR>\n" );
document.write( "There are n = 15 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( "15+13+15+15+14+12+12+14+14+15+14+15+16+13+13 = 210\r<BR>\n" );
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document.write( "Divide that sum by n = 15 to compute the mean<BR>\n" );
document.write( "mean = (sum of values)/n<BR>\n" );
document.write( "mean = (210)/15<BR>\n" );
document.write( "mean = 14<BR>\n" );
document.write( "This is the sample mean 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. We want the sample standard deviation.<BR>\n" );
document.write( "You should find the sample standard deviation is approximately s = 1.195229\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 = 1.195229/sqrt(15)<BR>\n" );
document.write( "SE = 0.3086068007936<BR>\n" );
document.write( "SE = 0.308607<BR>\n" );
document.write( "This value is approximate.\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 = (14 - 15)/0.308607<BR>\n" );
document.write( "t = -3.24036719841092<BR>\n" );
document.write( "t = -3.24\r<BR>\n" );
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document.write( "The sample size is n = 15<BR>\n" );
document.write( "The degrees of freedom (df) is<BR>\n" );
document.write( "df = n-1<BR>\n" );
document.write( "df = 15-1<BR>\n" );
document.write( "df = 14<BR>\n" );
document.write( "Use a stats calculator, spreadsheet, or similar to find the p-value is roughly 0.0059<BR>\n" );
document.write( "Since we're doing a two-tailed test, we find the area to the left of t = -3.24 and then double that result\r<BR>\n" );
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document.write( "The p-value being smaller than alpha = 0.05 tells us to reject the null and conclude that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=mu+%3C%3E+15 ALIGN=MIDDLE  ALT=\"mu+%3C%3E+15\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the case.\r<BR>\n" );
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document.write( "Therefore we conclude that the number of paintings owned by teachers is different from 15 paintings.<BR>\n" );
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