document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1181389>Question 1181389</A>: <I><SPAN STYLE=\"word-break: break-all;\"> At the Foremost State Bank the average savings account balance in 2012 was $1100. A random sample of 38 savings account balanes for 2013 yielded a mean of $1800 with a standard deviation of $2800. At the α = 0.10 significance level test the claim that the mean savings account balance in 2013 is different from the mean savings account balance in 2012.\r<BR>\n" );
document.write( "\n" );
document.write( "(a) Identify the correct alternative hypothesis.\r<BR>\n" );
document.write( "\n" );
document.write( "μ _____ $1100\r<BR>\n" );
document.write( "\n" );
document.write( "(b) The test statistic value is __________.  (Round your answer to two decimal places.)\r<BR>\n" );
document.write( "\n" );
document.write( "(c) Using the critical value approach (traditional method), the critical value is ____________ . (Round your answer to three decimal places.)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(d) Based on your answers above, do you <BR>\n" );
document.write( "Reject the  H^0 <BR>\n" );
document.write( "Fail to reject the  H^0<BR>\n" );
document.write( " \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(e) Explain your decision about the claim. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850037 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=850037\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's a breakdown of the solution:\r<BR>\n" );
document.write( "\n" );
document.write( "**(a) Alternative Hypothesis:**\r<BR>\n" );
document.write( "\n" );
document.write( "The claim is that the mean savings account balance in 2013 is *different* from the mean savings account balance in 2012.  \"Different\" means *not equal to*.  Therefore, the correct alternative hypothesis is:\r<BR>\n" );
document.write( "\n" );
document.write( "μ **≠** $1100\r<BR>\n" );
document.write( "\n" );
document.write( "**(b) Test Statistic:**\r<BR>\n" );
document.write( "\n" );
document.write( "We are given:\r<BR>\n" );
document.write( "\n" );
document.write( "* Sample mean (x̄) = $1800<BR>\n" );
document.write( "* Population mean (μ) = $1100<BR>\n" );
document.write( "* Sample size (n) = 38<BR>\n" );
document.write( "* Sample standard deviation (s) = $2800\r<BR>\n" );
document.write( "\n" );
document.write( "Since the sample size is greater than 30, we use a t-test. The test statistic is:\r<BR>\n" );
document.write( "\n" );
document.write( "t = (x̄ - μ) / (s / √n)<BR>\n" );
document.write( "t = (1800 - 1100) / (2800 / √38)<BR>\n" );
document.write( "t = 700 / (2800 / 6.1644)<BR>\n" );
document.write( "t = 700 / 454.16<BR>\n" );
document.write( "t ≈ 1.54\r<BR>\n" );
document.write( "\n" );
document.write( "The test statistic value is approximately **1.54**.\r<BR>\n" );
document.write( "\n" );
document.write( "**(c) Critical Value:**\r<BR>\n" );
document.write( "\n" );
document.write( "We're using a two-tailed test (because H₁ is ≠) with α = 0.10. Degrees of freedom (df) = n - 1 = 38 - 1 = 37.\r<BR>\n" );
document.write( "\n" );
document.write( "For a two-tailed test with α = 0.10 and df = 37, you'd consult a t-table or calculator.  You're looking for the t-value that cuts off 0.05 in *each* tail (since it's two-tailed).\r<BR>\n" );
document.write( "\n" );
document.write( "The critical values are approximately **±1.687**.\r<BR>\n" );
document.write( "\n" );
document.write( "**(d) Decision:**\r<BR>\n" );
document.write( "\n" );
document.write( "Our test statistic (1.54) falls *between* -1.687 and +1.687.  It's *not* in the rejection region (the tails).  Therefore, we *fail to reject* the null hypothesis.\r<BR>\n" );
document.write( "\n" );
document.write( "**Fail to reject the H₀**\r<BR>\n" );
document.write( "\n" );
document.write( "**(e) Explanation:**\r<BR>\n" );
document.write( "\n" );
document.write( "Because our calculated test statistic (1.54) falls within the non-rejection region (between the critical values), we do not have sufficient evidence at the α = 0.10 significance level to reject the null hypothesis.  In simpler terms, even though the sample mean is higher ($1800 vs $1100), there's too much variability in the sample (indicated by the large standard deviation) to confidently say that the true population mean for 2013 is *different* from $1100. The difference we see in the sample could reasonably be due to random chance.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );