document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1205286>Question 1205286</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Scarlett and Heather, the owners of an upscale restaurant in Dayton, Ohio, want to study the dining charac- teristics of their customers. They decide to focus on two variables: the amount of money spent by customers and<BR>\n" );
document.write( "whether customers order dessert. The results from a sample of 60 customers are as follows:<BR>\n" );
document.write( "Amount spent:sample mean,X = $38.54/customer and standard deviation of S = $7.62.<BR>\n" );
document.write( "18 customers purchased dessert out of 60 customers.<BR>\n" );
document.write( "a. Construct a 95% confidence interval estimate for the population mean amount spent per customer in the restaurant.<BR>\n" );
document.write( "b. Construct a 97% confidence interval estimate for the population proportion of customers who purchase dessert.<BR>\n" );
document.write( "Jack, the owner of a competing restaurant, wants to conduct a similar survey in his restaurant. Jack does not have access to the information that Scarlett and Heather have obtained from the survey they conducted. Answer the following questions:<BR>\n" );
document.write( "c. What sample size is needed to have 95% confidence of estimating the population mean amount spent in her restaurant with a margin of error of$1.50, assuming that the standard deviation is estimated to be $8?<BR>\n" );
document.write( "d. How many customers need to be selected to have 92% confidence of estimating the population proportion of customers who purchase dessert with a margin of error 0.04?<BR>\n" );
document.write( "e. Based on your answers to (c) and (d), how large a sample should Jack take? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #842210 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=842210\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> for scarlett and heather:<BR>\n" );
document.write( "sample mean for amount spent = 38.54<BR>\n" );
document.write( "sample standard deviation for amount spent = 7.62<BR>\n" );
document.write( "sample size is 60.<BR>\n" );
document.write( "18 out of 60 chose dessert.\r<BR>\n" );
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document.write( "since we are working off the sample mean and sample standard deviation, we need to use the t-score rather than the z-score.\r<BR>\n" );
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document.write( "a. Construct a 95% confidence interval estimate for the population mean amount spent per customer in the restaurant.\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "standard error = standard deviation / square root of sample size = 7.62/sqrt(60) equals .983738 rounded to 6 decimal places.\r<BR>\n" );
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document.write( "critical t-score at 95% confidence interval with 59 degrees of freedom is equal to t = plus or minus 2.0.\r<BR>\n" );
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document.write( "on the high side of the 95% confidence interval, the formula of t = (x-m)/s becomes 2.0 = (x-38.54)/.983738.<BR>\n" );
document.write( "solve for x to get x = 2.0 * .983738 + 38.54 = 40.5075 rounded to 4 decimal places.\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "on the low side of the 95% confidence interval, the formula of t = (x-m)/s becomes -2.0 = (x-38.54)/.983738.<BR>\n" );
document.write( "solve for x to get x = -2.0 * .983738 + 38.54 = 36.5725 rounded to 4 decimal places.\r<BR>\n" );
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document.write( "your 95% confidence interval is from 36.5725 to 40.5075.\r<BR>\n" );
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document.write( "b. Construct a 97% confidence interval estimate for the population proportion of customers who purchase dessert.\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "p = 18/60 = .3<BR>\n" );
document.write( "q = 1-p = .7<BR>\n" );
document.write( "standard error = sqrt(p*q/n)<BR>\n" );
document.write( "p = .3<BR>\n" );
document.write( "q = .7<BR>\n" );
document.write( "n = 60<BR>\n" );
document.write( "formula becomes standard error = sqrt(.3*.7/60) = .059161 rounded to 6 decimal places.\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "critical t-score with 59 degrees of freedom at 97% confidence interval becomes t = plus or minus 2.2238 rounded to 4 decimal places.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "on the high side of the confidence interval, t = (x-m)/s becomes 2.2238 = (x-.3) / .059161.<BR>\n" );
document.write( "solve for x to get x = 2.2238 * .059161 + .3 = .4316 rounded to 4 decimal places.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "on the low side of the confidence interval, t = (x-m)/s becomes -2.2238 = (x-.3) / .059161.<BR>\n" );
document.write( "solve for x to get x = -2.2238 * .059161 + .3 = .1684 rounded to 4 decimal places.\r<BR>\n" );
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document.write( "your 97% confidence interval is equal to .1684 to .4316\r<BR>\n" );
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document.write( "i believe part c and d will require the z-score.<BR>\n" );
document.write( "i'm beginning to think that part a and b also needed to assume the z-score.<BR>\n" );
document.write( "i can redo part a and b with z-scores if that's what you need.<BR>\n" );
document.write( "i'll leave part and b as is unless you tell me you need z-scores rather than t -scores.\r<BR>\n" );
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document.write( "for part c and beyond i'll assume z-scores.\r<BR>\n" );
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document.write( "here goes part c and beyond.\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "Jack, the owner of a competing restaurant, wants to conduct a similar survey in his restaurant. Jack does not have access to the information that Scarlett and Heather have obtained from the survey they conducted. Answer the following questions:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "c. What sample size is needed to have 95% confidence of estimating the population mean amount spent in her restaurant with a margin of error of$1.50, assuming that the standard deviation is estimated to be $8?\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "critical z-score at 95% confidence interval = plus or minus z = 1.96.<BR>\n" );
document.write( "formula used is z = (x-m)/s.<BR>\n" );
document.write( "(x-m) is the margin of error which is equal to 1.5\r<BR>\n" );
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document.write( "on the high side of the confidence interval, you get z = (x-m)/s which becomes 1.96 = 1.5 / s<BR>\n" );
document.write( "solve for s to get s = 1.5/1.96.<BR>\n" );
document.write( "since s = standard deviation / sqrt(sample size) and since standard deviation is equal to 8, then you get 8/sqrt(sample size) = 1.5/1.96<BR>\n" );
document.write( "solve for sqrt(sample size) to get sqrt(sample size) = 8 * 1.96 / 1.5 = 10.45333.....<BR>\n" );
document.write( "solve for sample size to get sample size = that squared = 109.27....<BR>\n" );
document.write( "since sample size has to be an integer, then round up to sample size to 110.<BR>\n" );
document.write( "standard error becomes 8/sqrt(110) = .762770 rounded to 6 decimal places.\r<BR>\n" );
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document.write( "on the high side of the confidence interval, the z-score formula becomes 1.96 = (x-m) / .762770 rounded to 6 decimal places.<BR>\n" );
document.write( "solve for (x-m) to get (x-m) = 1.96 * .762770 = 1.495 rounded to 3 decimal places.<BR>\n" );
document.write( "that's less than 1.5 as required.\r<BR>\n" );
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document.write( "on the low side of the confidence interval, the z-score formula becomes -1.96 = (x-m) / A.<BR>\n" );
document.write( "solve for (x-m) to get (x-m) = -1.96 * .762770 = -1.495 rounded to 3 decimal places.<BR>\n" );
document.write( "that's greater than -1.5 as required.\r<BR>\n" );
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document.write( "the mean can be anything and you will get a margin of error of less than 1.5.<BR>\n" );
document.write( "for example, assume the mean is 38.54.<BR>\n" );
document.write( "your 95% confidence interval will be 38.54 - 1.495 to 38.5 + 1.495 which is equal to 37.045 to 40.035.\r<BR>\n" );
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document.write( "here's what it looks like on a z-score calculator.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2023/122001.jpg\">\r<BR>\n" );
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document.write( "d. How many customers need to be selected to have 92% confidence of estimating the population proportion of customers who purchase dessert with a margin of error 0.04?\r<BR>\n" );
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document.write( "critical z-score for 92% confidence interval is equal to z = plus or minus 1.750686071.<BR>\n" );
document.write( "we'll round that to 1.7507.<BR>\n" );
document.write( "z-score formula is z = (x-m)/s<BR>\n" );
document.write( "(x-m) is the margin of error which is assumed to be .04.<BR>\n" );
document.write( "z-score formula on the high side of the confidence interval of z = (x-m)/s which becomes 1.7507 = (x-m)/s<BR>\n" );
document.write( "since (x-m) is assumed to be .04, the formula becomes 1.7507 = .04/s<BR>\n" );
document.write( "solve for s to get s = .04/1.7507 = .022848 rounded to 6 decimal places.\r<BR>\n" );
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document.write( "in a proportion type study, the standard error is equal to sqrt(p * q / n)<BR>\n" );
document.write( "since s is the standard error, then .022848 = sqrt(p * q / n)<BR>\n" );
document.write( "since the population proportion is assumed to be .3, then we get:<BR>\n" );
document.write( ".022848 = sqrt(.3*.7/n).<BR>\n" );
document.write( "square both sides of that to get:<BR>\n" );
document.write( ".022848^2 = .3*.7/n<BR>\n" );
document.write( "solve for n to get:<BR>\n" );
document.write( "n = .3*.7/.022848^2 = 402.27...<BR>\n" );
document.write( "since n has to be an integer, then n is equal to the next higher integer = 403.<BR>\n" );
document.write( "when n = 403, the standard error becomes sqrt(.3*.7/403) = .022827 rounded to 6 decimal places.\r<BR>\n" );
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document.write( "the high side of the confidence interval formula becomes 1.7507 = (x-m) / .022827.<BR>\n" );
document.write( "solve for (x-m) to get (x-m) = 1.7507 * .022827 = .03996 which is less than .04 as required.<BR>\n" );
document.write( "it won't be exactly .04 unless the sample size is not an integer.<BR>\n" );
document.write( "it's the largest the margin of error can be when the margin of error has to be less than .04.\r<BR>\n" );
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document.write( "here's what it looks like on a z-score calculator.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2023/122002.jpg\">\r<BR>\n" );
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document.write( "e. Based on your answers to (c) and (d), how large a sample should Jack take?\r<BR>\n" );
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document.write( "he needs to have the margin of error less than 1.5 for part c and less than .04 for part d.\r<BR>\n" );
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document.write( "he would need to use the larger sample size of 403 for both, if he only has 1 choice.<BR>\n" );
document.write( "that way, the margin of error will be less than 1.5 for part c and less than .04 for part d.\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );