document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1196535>Question 1196535</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A marketing research company is estimating which of two soft drinks college students prefer. A random sample of 100 college students produced the following 95% confidence interval for the proportion of college students who prefer drink A: (0.262, 0.622). What would happen to the confidence interval if the sample size were changed to 1 000?\r<BR>\n" );
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document.write( "The interval would get narrower.\r<BR>\n" );
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document.write( "It is impossible to tell until the interval is constructed.\r<BR>\n" );
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document.write( "The interval would get wider.\r<BR>\n" );
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document.write( "There would be no change in the width of the interva </SPAN>\n" );
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document.write( "Answer: <font color=red>A) The interval would get narrower.</font>\r<BR>\n" );
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document.write( "Reason:\r<BR>\n" );
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document.write( "When dealing with proportions, the margin of error formula is<BR>\n" );
document.write( "E = z*sqrt(phat*(1-phat)/n)<BR>\n" );
document.write( "where,<BR>\n" );
document.write( "z = critical value based on the confidence level<BR>\n" );
document.write( "phat = sample proportion<BR>\n" );
document.write( "n = sample size\r<BR>\n" );
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document.write( "If we fix z and phat to be constants, and let n vary, then E will decrease as n increases.\r<BR>\n" );
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document.write( "n goes up ---> E goes down<BR>\n" );
document.write( "and vice versa\r<BR>\n" );
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document.write( "The variables move in opposite directions of one another.\r<BR>\n" );
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document.write( "The margin of error getting smaller is an indication that we're narrowing in on the true population proportion. <BR>\n" );
document.write( "This is to be expected: The larger the sample, the better a sense we have about the true population proportion. <BR>\n" );
document.write( "There is less guesswork on what the population proportion is.\r<BR>\n" );
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document.write( "Think of it like fishing for an elusive fish. <BR>\n" );
document.write( "The more information you know, the smaller the net is needed to catch it.<BR>\n" );
document.write( "The less info you know, the net will have to be larger.<BR>\n" );
document.write( "The fishing net represents the confidence interval. \r<BR>\n" );
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document.write( "Let's look at a numeric example.<BR>\n" );
document.write( "I'll select a 95% confidence level and have phat = 0.50<BR>\n" );
document.write( "Feel free to use whatever phat value you prefer, as long as 0 < phat < 1<BR>\n" );
document.write( "Whatever you select, have it be fixed to a constant.\r<BR>\n" );
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document.write( "95% confidence means z = 1.96 approximately by use of a normal distribution table (aka Z table).<BR>\n" );
document.write( "Such tables are in the appendix section of your stats textbook. <BR>\n" );
document.write( "Or you can use an online one such as this<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf\">https://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf</a><BR>\n" );
document.write( "It says t table, but look at the very bottom row and it shows various Z values. The value 1.960 is right above the 95% confidence label.\r<BR>\n" );
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document.write( "Now consider a sample of n = 100<BR>\n" );
document.write( "Compute the margin of error<BR>\n" );
document.write( "E = z*sqrt(phat*(1-phat)/n)<BR>\n" );
document.write( "E = 1.96*sqrt(0.50*(1-0.50)/100)<BR>\n" );
document.write( "E = 0.098\r<BR>\n" );
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document.write( "Recompute with a larger sample of n = 1000<BR>\n" );
document.write( "Keep the other input values the same.<BR>\n" );
document.write( "E = z*sqrt(phat*(1-phat)/n)<BR>\n" );
document.write( "E = 1.96*sqrt(0.50*(1-0.50)/1000)<BR>\n" );
document.write( "E = 0.031\r<BR>\n" );
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document.write( "The results for each value of E are approximate.\r<BR>\n" );
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document.write( "The margin of error has gone from 0.098 to 0.031, which is a decrease.<BR>\n" );
document.write( "This is one example of the confidence interval shrinking as the sample size gets larger.\r<BR>\n" );
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document.write( "Similar question:<BR>\n" );
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