document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1176562>Question 1176562</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A random sample of 8 observations was drawn from a normal population. The sample mean and<BR>\n" );
document.write( "sample standard deviation are X = 40 and s = 10.<BR>\n" );
document.write( "a) Estimate the population mean with 95% confidence.<BR>\n" );
document.write( "b) Repeat part (a) assuming that you know that the population standard deviation is σ = 10. </SPAN>\n" );
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document.write( "Part (a)\r<BR>\n" );
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document.write( "The sample size is n = 8 and we don't know sigma, which is the population standard deviation. Instead, we have the sample standard deviation value s = 10. This sample statistic estimates the population parameter.\r<BR>\n" );
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document.write( "Because n > 30 is not true, and we don't know sigma, we must use the T distribution.\r<BR>\n" );
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document.write( "We have n-1 = 8-1 = 7 degrees of freedom. \r<BR>\n" );
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document.write( "Use a T table such as this one<BR>\n" );
document.write( "https://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf<BR>\n" );
document.write( "At the bottom of the table it shows the various confidence levels. Locate the 95% confidence level column. \r<BR>\n" );
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document.write( "Then mark the df = 7 row<BR>\n" );
document.write( "This is what you should have<BR>\n" );
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document.write( "We can see the value 2.365 is at the intersection of the row and column we highlighted.\r<BR>\n" );
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document.write( "The t critical value is roughly t = 2.365\r<BR>\n" );
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document.write( "Let's compute the lower bound L<BR>\n" );
document.write( "L = xbar - t*s/sqrt(n)<BR>\n" );
document.write( "L = 40 - 2.365*10/sqrt(8)<BR>\n" );
document.write( "L = 40 - 8.362<BR>\n" );
document.write( "L = 31.638<BR>\n" );
document.write( "L = 31.64\r<BR>\n" );
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document.write( "Now the upper bound U<BR>\n" );
document.write( "U = xbar + t*s/sqrt(n)<BR>\n" );
document.write( "U = 40 + 2.365*10/sqrt(8)<BR>\n" );
document.write( "U = 40 + 8.362<BR>\n" );
document.write( "U = 48.362<BR>\n" );
document.write( "U = 48.36\r<BR>\n" );
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document.write( "The 95% confidence interval in the form (L, U) is (31.64, 48.36)\r<BR>\n" );
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document.write( "<font color=red>Answer: (31.64, 48.36)</font>\r<BR>\n" );
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document.write( "We can write this in the form L < mu < U to say 31.64 < mu < 48.36<BR>\n" );
document.write( "This format is more descriptive in that it's more clear that we're estimating mu here. \r<BR>\n" );
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document.write( "Part (b)\r<BR>\n" );
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document.write( "Now we're told that sigma = 10, while everything else is kept the same.\r<BR>\n" );
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document.write( "Since we know sigma, we can use the Z distribution this time.\r<BR>\n" );
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document.write( "Using a Z table, the critical value is roughly z = 1.960 at 95% confidence.\r<BR>\n" );
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document.write( "Lower Bound<BR>\n" );
document.write( "L = xbar - z*sigma/sqrt(n)<BR>\n" );
document.write( "L = 40 - 1.960*10/sqrt(8)<BR>\n" );
document.write( "L = 40 - 6.930<BR>\n" );
document.write( "L = 33.07\r<BR>\n" );
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document.write( "Upper Bound<BR>\n" );
document.write( "U = xbar + z*sigma/sqrt(n)<BR>\n" );
document.write( "U = 40 + 1.960*10/sqrt(8)<BR>\n" );
document.write( "U = 40 + 6.930<BR>\n" );
document.write( "U = 46.93\r<BR>\n" );
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document.write( "As you can see, the format and structure of each formula is pretty much identical to the T distribution variety used in part (a). The only difference is that t has been replaced with z (so 2.365 is replaced with 1.960), and that we used sigma in place of s.\r<BR>\n" );
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document.write( "<font color=red>Answer: (33.07, 46.93)</font>\r<BR>\n" );
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document.write( "This is equivalent to saying 33.07 < mu < 46.93<BR>\n" );
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