document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1185508>Question 1185508</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Do you want to own your own candy store? Wow! With some interest in running your own business and a decent credit rating, you can probably get a bank loan on startup costs for franchises such as Candy Express, The Fudge Company, Karmel Corn, and Rocky Mountain Chocolate Factory. Startup costs (in thousands of dollars) for a random sample of candy stores are given below. Assume that the population of x values has an approximately normal distribution.<BR>\n" );
document.write( "96	178	126	94	75	94	116	100	85<BR>\n" );
document.write( "A button hyperlink to the SALT program that reads: Use SALT.<BR>\n" );
document.write( "(a) Use a calculator with mean and sample standard deviation keys to find the sample mean startup cost x and sample standard deviation s. (Round your answers to four decimal places.)<BR>\n" );
document.write( "x =	<BR>\n" );
document.write( " thousand dollars<BR>\n" );
document.write( "s =	<BR>\n" );
document.write( " thousand dollars\r<BR>\n" );
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document.write( "(b) Find a 90% confidence interval for the population average startup costs 𝜇 for candy store franchises. (Round your answers to one decimal place.)<BR>\n" );
document.write( "lower limit    	<BR>\n" );
document.write( " thousand dollars<BR>\n" );
document.write( "upper limit    	<BR>\n" );
document.write( " thousand dollars </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849692 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849692\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to calculate the sample mean, sample standard deviation, and the 90% confidence interval:\r<BR>\n" );
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document.write( "**(a) Sample Mean and Standard Deviation:**\r<BR>\n" );
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document.write( "1. **Sample Mean (x̄):**  Sum all the startup costs and divide by the number of stores (n = 9):\r<BR>\n" );
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document.write( "   x̄ = (96 + 178 + 126 + 94 + 75 + 94 + 116 + 100 + 85) / 9<BR>\n" );
document.write( "   x̄ = 964 / 9<BR>\n" );
document.write( "   x̄ ≈ 107.1111 thousand dollars\r<BR>\n" );
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document.write( "2. **Sample Standard Deviation (s):**\r<BR>\n" );
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document.write( "   First, calculate the squared differences from the mean for each value, sum them, divide by (n-1), and then take the square root:\r<BR>\n" );
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document.write( "   s = √[Σ(xi - x̄)² / (n - 1)]\r<BR>\n" );
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document.write( "   Here's a breakdown:\r<BR>\n" );
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document.write( "   * (96-107.11)² = 123.46<BR>\n" );
document.write( "   * (178-107.11)² = 4997.74<BR>\n" );
document.write( "   * (126-107.11)² = 356.66<BR>\n" );
document.write( "   * (94-107.11)² = 171.86<BR>\n" );
document.write( "   * (75-107.11)² = 1031.54<BR>\n" );
document.write( "   * (94-107.11)² = 171.86<BR>\n" );
document.write( "   * (116-107.11)² = 78.94<BR>\n" );
document.write( "   * (100-107.11)² = 50.54<BR>\n" );
document.write( "   * (85-107.11)² = 488.94\r<BR>\n" );
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document.write( "   Sum of squared differences: 7471.22<BR>\n" );
document.write( "   s = √(7471.22 / 8)<BR>\n" );
document.write( "   s ≈ √933.9025<BR>\n" );
document.write( "   s ≈ 30.56 thousand dollars\r<BR>\n" );
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document.write( "**(b) 90% Confidence Interval:**\r<BR>\n" );
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document.write( "Since the population standard deviation is unknown and the sample size is small (n < 30), we use a t-distribution.\r<BR>\n" );
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document.write( "1. **Degrees of Freedom (df):** df = n - 1 = 9 - 1 = 8\r<BR>\n" );
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document.write( "2. **Critical Value (t_c):** For a 90% confidence level and df = 8, look up the t-value in a t-table or use a calculator. The t_c ≈ 1.860.\r<BR>\n" );
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document.write( "3. **Margin of Error (E):**\r<BR>\n" );
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document.write( "   E = t_c * (s / √n)<BR>\n" );
document.write( "   E = 1.860 * (30.56 / √9)<BR>\n" );
document.write( "   E = 1.860 * (30.56 / 3)<BR>\n" );
document.write( "   E ≈ 19.01\r<BR>\n" );
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document.write( "4. **Confidence Interval:**\r<BR>\n" );
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document.write( "   Lower Limit = x̄ - E = 107.11 - 19.01 ≈ 88.1 thousand dollars<BR>\n" );
document.write( "   Upper Limit = x̄ + E = 107.11 + 19.01 ≈ 126.1 thousand dollars\r<BR>\n" );
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document.write( "Therefore:\r<BR>\n" );
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document.write( "* x̄ = 107.1111 thousand dollars<BR>\n" );
document.write( "* s = 30.5600 thousand dollars<BR>\n" );
document.write( "* Lower Limit = 88.1 thousand dollars<BR>\n" );
document.write( "* Upper Limit = 126.1 thousand dollars<BR>\n" );
document.write( " </SPAN>\n" );
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