document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1169988>Question 1169988</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi, I am in need of help! if you can help me solve that would be great!thank you\r<BR>\n" );
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document.write( "The mean gas mileage for a hybrid car is 57 miles per gallon. Suppose that the gasoline mileage is approximately normally distributed with a standard deviation of 3.5 miles per gallon.​ <BR>\n" );
document.write( "a) What proportion of hybrids gets over 62  miles per​ gallon?<BR>\n" );
document.write( "b) What proportion of hybrids gets 53 miles per gallon or​ less?<BR>\n" );
document.write( "c) what proportion of hybrids gets between 59 and 62 miles per​ gallon? <BR>\n" );
document.write( "d) What is the probability that a randomly selected hybrid gets less than 46 miles per​ gallon? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #794870 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=794870\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> you can use a calculator or you can use the normal distribution tables.\r<BR>\n" );
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document.write( "first step is to get the z-scores.\r<BR>\n" );
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document.write( "the formula for z-score is:\r<BR>\n" );
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document.write( "z = (x - m) / s\r<BR>\n" );
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document.write( "z is the z-score<BR>\n" );
document.write( "x i the raw score<BR>\n" );
document.write( "m is the mean<BR>\n" );
document.write( "s is the standard deviation\r<BR>\n" );
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document.write( "you are given that the mean is 57 and the standard deviation is 3.5\r<BR>\n" );
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document.write( "if the distribution is normal, then the mean, median, and mode coincide at the middle of the distribution.\r<BR>\n" );
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document.write( "the mean is the average<BR>\n" );
document.write( "the median is the point where half of the data falls below and half of the data falls above.<BR>\n" );
document.write( "the mode is is the most frequently occurring value.\r<BR>\n" );
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document.write( "here's a  description.<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://statistics.laerd.com/statistical-guides/measures-central-tendency-mean-mode-median.php\" target = \"_blank\">https://statistics.laerd.com/statistical-guides/measures-central-tendency-mean-mode-median.php</a>\r<BR>\n" );
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document.write( "the total area underneath the normal distribution curve is equal to 1.<BR>\n" );
document.write( "this id done by design.<BR>\n" );
document.write( "a point, on the normal distribution curve has a certain amount of the area to the left of it and to the right of it.\r<BR>\n" );
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document.write( "this area is expressed as a proportion or percent.\r<BR>\n" );
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document.write( "for example, the mean is exactly in the middle and, since the curve is symmetric, half the area is to the left and half the area is to the right.<BR>\n" );
document.write( "the right.\r<BR>\n" );
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document.write( "the z-score at the mean = 0.<BR>\n" );
document.write( "all z-scores are in relation to the mean.<BR>\n" );
document.write( "z-scores to the left of the mean are negative.<BR>\n" );
document.write( "z-scores to the right of the mean are positive.\r<BR>\n" );
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document.write( "the tables are set up to give you the area to the left of the z-score.<BR>\n" );
document.write( "if you want the area to the right of the z-score, you would subtract the area from 1.\r<BR>\n" );
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document.write( "for example:\r<BR>\n" );
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document.write( "a z-score of 1 gives you an area to the left of it as .84134 to the left of it.<BR>\n" );
document.write( "the area to the right of it would be equal to 1 minus .84134 = .15866.\r<BR>\n" );
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document.write( "since the z-scores are symmetric about the mean, then a z-score of - 1 would have an area of .15866 to the left of it and an area of 1 - .15866 = .84134 to the right of it.\r<BR>\n" );
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document.write( "the sum of the areas to the left of the z-score and to the right of the z-scores will always be equal to 1.\r<BR>\n" );
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document.write( "to use the z-score tables, you need to translate the raw score to the z-score.\r<BR>\n" );
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document.write( "here's a table where you can check that out for yourself.\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"https://www.math.arizona.edu/~rsims/ma464/standardnormaltable.pdf\" target = \"_blank\">https://www.math.arizona.edu/~rsims/ma464/standardnormaltable.pdf</a>\r<BR>\n" );
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document.write( "the z-score formula was mentioned above and is repeated here.\r<BR>\n" );
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document.write( "the formula is z = (x - m) / s\r<BR>\n" );
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document.write( "x is the raw score<BR>\n" );
document.write( "m is the mean<BR>\n" );
document.write( "s is the standard deviation.\r<BR>\n" );
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document.write( "you are given that the mean is 57 and the standard deviation is 3.5\r<BR>\n" );
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document.write( "your questions are below and will be answedred in turn.\r<BR>\n" );
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document.write( "a) What proportion of hybrids gets over 62 miles per​ gallon?\r<BR>\n" );
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document.write( "z = (62 - 57) / 3.5<BR>\n" );
document.write( "solve for z to get:<BR>\n" );
document.write( "z = 5/3.5 = 1.43 rounded to 2 decimal places.<BR>\n" );
document.write( "look in the table to get area to the left of that z-score = .92364.<BR>\n" );
document.write( "area to the right of it = 1 - .92364 = .07636.<BR>\n" );
document.write( "you are looking for the area to the right of it, therefore:<BR>\n" );
document.write( "the proportion of hybrids that get over 62 miles per gallon is .07636.\r<BR>\n" );
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document.write( "b) What proportion of hybrids gets 53 miles per gallon or​ less?\r<BR>\n" );
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document.write( "z = (53 - 57) / 3.5<BR>\n" );
document.write( "solve for z to get:<BR>\n" );
document.write( "z = -4/3.5 = -1.14 rounded to 2 decimal places.<BR>\n" );
document.write( "look in the table to get area to the left of that z-score = .12714.<BR>\n" );
document.write( "area to the right of it = 1 - .12714 = .87286.<BR>\n" );
document.write( "you are looking for the area to the left of it, therefore:<BR>\n" );
document.write( "the proportion of hybrids that get less than 53 miles per gallon is .12714.\r<BR>\n" );
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document.write( "c) what proportion of hybrids gets between 59 and 62 miles per​ gallon?\r<BR>\n" );
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document.write( "you need the low z-score and the high z-score.\r<BR>\n" );
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document.write( "for the low z-score:<BR>\n" );
document.write( "z = (59 - 57) / 3.5<BR>\n" );
document.write( "solve for z to get:<BR>\n" );
document.write( "z = 0.57 rounded to 2 decimal places.<BR>\n" );
document.write( "look in the table to get area to the left of that = .71566.<BR>\n" );
document.write( "area to the right of it = 1 - .71566 = .28434\r<BR>\n" );
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document.write( "for the high z-score:<BR>\n" );
document.write( "z = (62 - 57) / 3.5<BR>\n" );
document.write( "solve for z to get:<BR>\n" );
document.write( "z = 5/3.5 = 1.43 rounded to 2 decimal places.<BR>\n" );
document.write( "look in the table to get area to the left of that = .92364.<BR>\n" );
document.write( "area to the right of that = 1 - .92364 = .07636.\r<BR>\n" );
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document.write( "for this problem, you need the area to the left of both z-scores.<BR>\n" );
document.write( "you would subtract the smaller area from the larger area to get:<BR>\n" );
document.write( ".92364 - .71566 = .20798.<BR>\n" );
document.write( "the proportion of hybrids that get between 53 and 62 miles per gallon is .20798.\r<BR>\n" );
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document.write( "d) What is the probability that a randomly selected hybrid gets less than 46 miles per​ gallon?\r<BR>\n" );
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document.write( "z = (46 - 57) / 3.5<BR>\n" );
document.write( "solve for z to get:<BR>\n" );
document.write( "z = -11/3.5 = -3.14 rounded to 2 decimal places.<BR>\n" );
document.write( "look in the table to get area to the left of that = .00084<BR>\n" );
document.write( "area to the right of that = 1 - .00084 = .99916<BR>\n" );
document.write( "you want the area to the left of that.\r<BR>\n" );
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document.write( "the proportion of cars that get less than 46 miles per gallon is .00084.\r<BR>\n" );
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document.write( "that is also the probability that a randomly selected car will get less than 46 miles per gallon.\r<BR>\n" );
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document.write( "that should answer all your questions.<BR>\n" );
document.write( "some additional comments below.\r<BR>\n" );
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document.write( "proportion * 100 = percent<BR>\n" );
document.write( "so if the proportion of cars that get less than 46 miles per gallon is .00084, then the percent of cars that get less than 46 miles per gallon is .084%.\r<BR>\n" );
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document.write( "proportions and percents and probability are related.<BR>\n" );
document.write( "if the proportion of cars that gets less than 62 miles per gallon is .92364, then the percent of cars that get less than 62 miles per gallon is 92.364% and the probability that a randomly selected car will get less than 62 miles per gallon is .92364.\r<BR>\n" );
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document.write( "the tables are set up to give you the area to the left of the z-score.<BR>\n" );
document.write( "you look in the first column to find the z-score to the first decimal digit.<BR>\n" );
document.write( "you then look across the rows to get the z-score to the second decimal digit.\r<BR>\n" );
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document.write( "for example, to find the area to the left of a z-score of -3.56, you look down the first column to find a z-score of -3.5.<BR>\n" );
document.write( "you then look across that row to find the z-score under .06.<BR>\n" );
document.write( "the area that you find at the intersection of the row that -3.5 is on and the column that .06 is on is the area to the left of the z-score of -3.56.\r<BR>\n" );
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document.write( "here's a picture of what i mean.\r<BR>\n" );
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document.write( "the area to the left of a z-score of -3.56 is equal to .00019.\r<BR>\n" );
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document.write( "so far we've use the tables.<BR>\n" );
document.write( "however, there are calculators on the web that can help with this.<BR>\n" );
document.write( "one such calculator can be found at <a  rel=nofollow HREF= \"https://www.calculator.net/z-score-calculator.html?c1raw=3.075&c1mean=3.2&c1sd=.1&calctype=zscore&x=81&y=18\" target = \"blank\">https://www.calculator.net/z-score-calculator.html?c1raw=3.075&c1mean=3.2&c1sd=.1&calctype=zscore&x=81&y=18</a>\r<BR>\n" );
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document.write( "for example, area to the left of a z-score of 1.43 would be shown as .92364 in this calculator.<BR>\n" );
document.write( "this agrees with the number you got from the table.<BR>\n" );
document.write( "if you entered the raw score rather than the z-score, the calculator would have told you that the area to the left of the raw score was equal to .92344 and the z-score was 1.42857.\r<BR>\n" );
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document.write( "there are 3 sections to the table.<BR>\n" );
document.write( "learn to use those sections and they can help you as well.<BR>\n" );
document.write( "the page the calculator is on also shows you the z-score table.\r<BR>\n" );
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document.write( "all in all, a pretty decent piece of work that can help you once you learn how to use it properly.\r<BR>\n" );
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document.write( "here's a couple of results from the examples i used.\r<BR>\n" );
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document.write( "if you have any questions about any of this, just let mem know.\r<BR>\n" );
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document.write( "theo\r<BR>\n" );
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document.write( " </SPAN>\n" );
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