document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1196534>Question 1196534</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 confidence level were changed to 90%?\r<BR>\n" );
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document.write( "There would be no change in the width of the interval.\r<BR>\n" );
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document.write( "The interval would get narrower.\r<BR>\n" );
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document.write( "The interval would get wider.\r<BR>\n" );
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document.write( "It is impossible to tell until the 90% interval is constructed </SPAN>\n" );
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document.write( "Answer: <font color=red>B) The interval would get narrower.</font>\r<BR>\n" );
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document.write( "Explanation:\r<BR>\n" );
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document.write( "The margin of error formula for proportions 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( "We'll fix phat and n to be constant<BR>\n" );
document.write( "They won't change in value.\r<BR>\n" );
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document.write( "Let's say phat = 0.50 and n = 100\r<BR>\n" );
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document.write( "At 95% confidence, the z critical value is roughly 1.96<BR>\n" );
document.write( "At 90% confidence, the z critical value is roughly 1.645<BR>\n" );
document.write( "Use a table like 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( "to get those values. Look at the bottom row labeled \"Z\".\r<BR>\n" );
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document.write( "Compute the value of E for the 95% confidence level<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( "Now do the same for the 90% confidence level. <BR>\n" );
document.write( "Keep phat and n the same from earlier.<BR>\n" );
document.write( "E = z*sqrt(phat*(1-phat)/n)<BR>\n" );
document.write( "E = 1.645*sqrt(0.50*(1-0.50)/100)<BR>\n" );
document.write( "E = 0.08225\r<BR>\n" );
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document.write( "The margin of error has gone from 0.098 to 0.08225<BR>\n" );
document.write( "The margin of error decreases when the confidence level decreases. \r<BR>\n" );
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document.write( "This makes the confidence interval get more narrow.<BR>\n" );
document.write( "The confidence interval has the width of 2*E, so we can think of E as the radius of the confidence interval.\r<BR>\n" );
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document.write( "Here's an alternative way to think about it:\r<BR>\n" );
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document.write( "Let's say your friend drew out a number line, and marked it from 1 to 10 including both endpoints. <BR>\n" );
document.write( "Then your friend picked one of the numbers on that number line.<BR>\n" );
document.write( "Your goal is to guess their number.\r<BR>\n" );
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document.write( "You are 100% confident it is somewhere between 1 and 10. <BR>\n" );
document.write( "This is the largest confidence interval possible, and hence the widest width possible along the number line.\r<BR>\n" );
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document.write( "If you lowered the confidence down to 90%, then you could have a subinterval of say from 2 to 10, or something along those lines. <BR>\n" );
document.write( "The friend's number may be in this range, but you aren't fully certain. \r<BR>\n" );
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document.write( "Then an 80% confidence interval could be going from 3 to 10 as one possible subinterval. The level of uncertainty has dropped further.\r<BR>\n" );
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document.write( "Each time the confidence decreases, the interval shrinks.\r<BR>\n" );
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document.write( "Similar question:<BR>\n" );
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