document.write( "Question 1200018: The average sales price of a single-family house in the United States is $290,600. You randomly select 12 single-family houses. What is the probability that the mean sales price is more than $265,000? Assume that the sales prices are normally distributed with a standard deviation of $36,000. (Adapted from The U.S. Commerce Department) a. Use the Central Limit Theorem to find and and sketch the sampling distribution of the sample means. b. Find the z-score that corresponds to $265,000. c. Find the cumulative area that corresponds to the z-score and calculate the probability. d. Interpret the results. \n" ); document.write( "
Algebra.Com's Answer #834025 by Theo(13342)\"\" \"About 
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the mean is 290,600
\n" ); document.write( "the standard deviation is 36,000
\n" ); document.write( "sample size is 12.
\n" ); document.write( "standard error is equal to standard deviation divided by square root of sample size = 36,000 / sqrt(12)
\n" ); document.write( "z-score formula is z = (x - m) / s
\n" ); document.write( "z is the z-scoe
\n" ); document.write( "x is the raw score
\n" ); document.write( "m is the mean
\n" ); document.write( "s is the standard error
\n" ); document.write( "when x = 265,000, the formula becomes:
\n" ); document.write( "z = (265,000 - 290,000) / (36,000 / sqrt(12)).
\n" ); document.write( "solve for z to get:
\n" ); document.write( "z = -2.405626122
\n" ); document.write( "area to the left of that z-score = .0080723832
\n" ); document.write( "area to the right of that z-score = 1 minus that = .9919276168
\n" ); document.write( "this tells you that the probability of getting a sample of 12 houses where the mean price is less than 265,000 is equal to .0080723832 and the probability of getting a sample of 12 house where the mean price is greater than 265,000 is equal to .9919276168
\n" ); document.write( "if you were to sketch the graph of the normal probability curve, you would get:\r
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\n" ); document.write( "\n" ); document.write( "with z-scores:\r
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\n" ); document.write( "\n" ); document.write( "with raw scores:\r
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\n" ); document.write( "\n" ); document.write( "when you're working with z-scores, the meanis 0 and the standard deviation is 1.\r
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\n" ); document.write( "\n" ); document.write( "then you're working with raw scores, the mean is the population mean and the standard deviation is the standard error.\r
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\n" ); document.write( "\n" ); document.write( "with sample means, you use the standad error which is the standard deviation divided by the square root of the sample size.\r
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\n" ); document.write( "\n" ); document.write( "with z-scores, the area to the left of the z-score is the probability you will find a z-score less than the specified z-score and the area to the right of the z-score is the probability you will find a z-score more than the specified z-score.\r
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\n" ); document.write( "\n" ); document.write( "with raw scores, the area to the left of the raw score is the probability yu will find a raw score less than the specified raw score and the area to the right of the raw score is the probability you will find a raw score more than the specified raw score.\r
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\n" ); document.write( "\n" ); document.write( "the calculator is capable of doing it either way.\r
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\n" ); document.write( "\n" ); document.write( "the calculator can be found at https://davidmlane.com/hyperstat/z_table.html
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