document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1110607>Question 1110607</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Allen's hummingbird (Selasphorus sasin) has been studied by zoologist Bill Alther.† Suppose a small group of 15 Allen's hummingbirds has been under study in Arizona. The average weight for these birds is x = 3.15 grams. Based on previous studies, we can assume that the weights of Allen's hummingbirds have a normal distribution, with &#963; = 0.36 gram.<BR>\n" );
document.write( "(a) Find an 80% confidence interval for the average weights of Allen's hummingbirds in the study region. What is the margin of error? (Round your answers to two decimal places.)<BR>\n" );
document.write( "lower limit    	 <BR>\n" );
document.write( "upper limit    	 <BR>\n" );
document.write( "margin of error    	 \r<BR>\n" );
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document.write( "(b) What conditions are necessary for your calculations? (Select all that apply.)<BR>\n" );
document.write( "&#963; is unknown<BR>\n" );
document.write( "uniform distribution of weights<BR>\n" );
document.write( "&#963; is known<BR>\n" );
document.write( "n is large<BR>\n" );
document.write( "normal distribution of weights\r<BR>\n" );
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document.write( "(c) Interpret your results in the context of this problem.<BR>\n" );
document.write( "The probability that this interval contains the true average weight of Allen's hummingbirds is 0.80.<BR>\n" );
document.write( "There is an 80% chance that the interval is one of the intervals containing the true average weight of Allen's hummingbirds in this region.    <BR>\n" );
document.write( "The probability that this interval contains the true average weight of Allen's hummingbirds is 0.20.<BR>\n" );
document.write( "There is a 20% chance that the interval is one of the intervals containing the true average weight of Allen's hummingbirds in this region.<BR>\n" );
document.write( "The probability to the true average weight of Allen's hummingbirds is equal to the sample mean.\r<BR>\n" );
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document.write( "(d) Find the sample size necessary for an 80% confidence level with a maximal margin of error E = 0.10 for the mean weights of the hummingbirds. (Round up to the nearest whole number.) <BR>\n" );
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document.write( " hummingbirds </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #725586 by </B><A HREF=/tutors/aboutme.mpl?userid=Boreal><B>Boreal(15235)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Boreal><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=Boreal><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=725586\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The width of the interval is z(0.90)*sigma/sqrt(n)<BR>\n" );
document.write( "this is 1.28*0.36/sqrt(15)=0.1190 or 0.12<BR>\n" );
document.write( "The interval is (3.03, 3.27) units gms.\r<BR>\n" );
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document.write( "I need sigma is known and normal distribution.  They go together, and when I have them, n can be anything, so it is not required to be large.\r<BR>\n" );
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document.write( "The second choice that there is an 80% chance the interval is one of those... I like to say that if I construct 100 similar intervals for this sample size, 80 of them will contain the parameter.  I don't know which 80, so the probability can be used.  The important concept is not that there is an 80% chance the mean is in this interval.  It either is or it isn't, which is why we use confidence and not probability.\r<BR>\n" );
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document.write( "The last part requires 1.28*0.36/sqrt(n) to be < 0.1<BR>\n" );
document.write( "0.2123<.01 n, squaring everything<BR>\n" );
document.write( "21.23 is the sample size required, or n=22 </SPAN>\n" );
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