document.write( "Question 1199108: It takes runners on average 289 minutes to run the marathon. The standard deviation of the times to run the marathon is known to be 26 minutes.
\n" ); document.write( "Drink ZXZ manufacturer claims that drinking ZXZ helps runners to run the
\n" ); document.write( "marathon faster. To test that claim, we took 30 runners, and gave them to drink
\n" ); document.write( "ZXZ before they ran the marathon. In our sample, the average time to run the
\n" ); document.write( "marathon was 281 minutes.
\n" ); document.write( "Part a) What is H0 and what is H1 in this test?
\n" ); document.write( "Part b) Do we have enough evidence to reject H0 and accept H1? \n" ); document.write( "
Algebra.Com's Answer #832804 by Theo(13342)\"\" \"About 
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the population mean is assumed to be 289 minutes and the population standard deviation is assumed to be 26 minutes.\r
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\n" ); document.write( "\n" ); document.write( "the sample size is 30 runners.
\n" ); document.write( "the sample mean is 281 minutes.\r
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\n" ); document.write( "\n" ); document.write( "H0 says that the average mean is what the assumed population mean is (289)
\n" ); document.write( "H1 says that the average mean is not 289.\r
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\n" ); document.write( "\n" ); document.write( "since you are looking at the mean of the sample, you use the standard error rather than the standard deviation.
\n" ); document.write( "standard error = standard deviation / square root of sample size = 26 / sqrt(30) = 4.7469 rounded to 4 decimal places.\r
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\n" ); document.write( "\n" ); document.write( "use the z-score formula to find the z-score.
\n" ); document.write( "the z-score formula is z = (x - m) / s
\n" ); document.write( "in this case, .....
\n" ); document.write( "z is the z-score
\n" ); document.write( "x is the mean of the sample
\n" ); document.write( "m is the assumed population mean.
\n" ); document.write( "s is the standard error.\r
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\n" ); document.write( "\n" ); document.write( "you get z = (281 - 289) / 4.7469 = -1.6893 rounded to 4 decimal places.\r
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\n" ); document.write( "\n" ); document.write( "the area to the left of that z-score is equal to .0456 rounded to 4 decimal places.\r
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\n" ); document.write( "\n" ); document.write( "since your test was for not equal, you would have a two tailed confidence interval.
\n" ); document.write( "at 95% confidence interval, the tail on each end would be 2.5% = .025.
\n" ); document.write( "since the test tail on the left was .0456, this is greater thaan the critical tail at .025.
\n" ); document.write( "consequently you would assume that you did not have enough information to conclude that the population mean was not equal to 289.
\n" ); document.write( "the difference would be assumed to be caused by random variation in the sample mean.\r
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\n" ); document.write( "\n" ); document.write( "here's what the test results would look like on the z-score calculator at https://davidmlane.com/hyperstat/z_table.html\r
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