document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=629065>Question 629065</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose that we will randomly select a sample of 64 measurements from a population having a mean equal to 20 and a standard deviation equal to 4.<BR>\n" );
document.write( "a. Describe the shape of the sampling distribution of the sample mean x bar. Do we need to make any assumptions about the shape of the population? Why or why not?<BR>\n" );
document.write( "b. Find the mean and the standard deviation of the sampling distribution of the sample mean x bar. <BR>\n" );
document.write( "c. Calculate the probability that we will obtain a sample mean greater than 21; that is, calculate P(x bar>21). Hint: Find the z value corresponding to 21 by using ?, and ? because we wish to calculate a probability about x bar. Then sketch the sampling distribution and the probability. <BR>\n" );
document.write( "d. Calculate the probability that we will obtain a sample mean less than 19.385; that is, calculate P(x bar <19.385).  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #396072 by </B><A HREF=/tutors/aboutme.mpl?userid=stanbon><B>stanbon(75887)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=stanbon><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=396072\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Suppose that we will randomly select a sample of 64 measurements from a population having a mean equal to 20 and a standard deviation equal to 4.<BR>\n" );
document.write( "a. Describe the shape of the sampling distribution of the sample mean x-bar.<BR>\n" );
document.write( "Approaching normal.<BR>\n" );
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document.write( "Do we need to make any assumptions about the shape of the population? Why or why not?<BR>\n" );
document.write( "Ans: Independence of the data.<BR>\n" );
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document.write( "b. Find the mean and the standard deviation of the sampling distribution of the sample mean x-bar.<BR>\n" );
document.write( "mean of the sample means = mean of the population<BR>\n" );
document.write( "std of the sample means = s/sqrt(n)<BR>\n" );
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document.write( "c. Calculate the probability that we will obtain a sample mean greater than 21; that is, calculate P(x bar>21). Hint: Find the z value corresponding to 21 by using ?, and ? because we wish to calculate a probability about x-bar. Then sketch the sampling distribution and the probability.<BR>\n" );
document.write( "z(21) = (21-20)/(4/sqrt(64)) = 1/(1/2) = 2<BR>\n" );
document.write( "P(x-bar < 21) = P(z < 2) = 0.9772<BR>\n" );
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document.write( "d. Calculate the probability that we will obtain a sample mean less than 19.385; that is, calculate P(x bar <19.385).<BR>\n" );
document.write( "z(19.385) = (19.385-20)/[4/8] = 0.385/0.5 = 0.77<BR>\n" );
document.write( "P(x bar <19.385) = P(z < 0.77) = 0.7794<BR>\n" );
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document.write( "Cheers,<BR>\n" );
document.write( "Stan H.<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );