document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1039865>Question 1039865</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A study is designed to investigate whether there is a difference in response to various treatments in patients with rheumatoid arthritis. The outcome is patient's self-reported effect of treatment. The data are shown below. Is there a statistically significant difference in the proportions of patients who show improvement between treatments 1 and 2. Apply the test at a 5% level of significance. \r<BR>\n" );
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document.write( "Treatment 1: 22 symptoms worsened-14 no effect-14 symptoms improved.<BR>\n" );
document.write( "Treatment 2: 14 symptoms worsened-15 no effect-21 symptoms improved.\r<BR>\n" );
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document.write( "I need the critical value and the computed statistic. I got 17.3 and 26.7 which are both wrong but I don't know why.\r<BR>\n" );
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document.write( "The second half of this question is: based on the above figures (which I got wrong), Which of the following is or are true statement(s)?\r<BR>\n" );
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document.write( "A. There is significant evidence, alpha 0.05, to show that there is a difference in the proportion of patients who show improvement between treatments 1 and 2.<BR>\n" );
document.write( "B. There is Not significant evidence, alpha 0.05 to show that there is a difference in the proportion of patients who show improvements between treatments 1 and 2.<BR>\n" );
document.write( "C. There is significant evidence, alpha 0.05, to show that there is NO difference in the proportions of patients who show improvements between treatments 1 and 2.<BR>\n" );
document.write( "D. Both A and C are correct.\r<BR>\n" );
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document.write( "I keep getting this whole problem wrong. Please help if you can as I need to understand how to do these and judge whether they are significant or not. My book in not really that helpful.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #654649 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=654649\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3>\r<BR>\n" );
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document.write( "Variables used<BR>\n" );
document.write( "x1 = number of people who improved under treatment 1<BR>\n" );
document.write( "p1 = population proportion of patients who improve under treatment 1<BR>\n" );
document.write( "phat1 = sample proportion of patients who improve under treatment 1<BR>\n" );
document.write( "n1 = sample size of treatment 1\r<BR>\n" );
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document.write( "x2 = number of people who improved under treatment 2<BR>\n" );
document.write( "p2 = population proportion of patients who improve under treatment 2<BR>\n" );
document.write( "phat2 = sample proportion of patients who improve under treatment 2<BR>\n" );
document.write( "n2 = sample size of treatment 2\r<BR>\n" );
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document.write( "qhat = average of phat1 and phat2\r<BR>\n" );
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document.write( "alpha = 0.05 is the significance level\r<BR>\n" );
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document.write( "The hypothesis are<BR>\n" );
document.write( "Null: <BR>\n" );
document.write( "H0: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p1+=+p2 ALIGN=MIDDLE  ALT=\"p1+=+p2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Alternate:<BR>\n" );
document.write( "H1: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p1+%3C%3E+p2 ALIGN=MIDDLE  ALT=\"p1+%3C%3E+p2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "This is a two tailed test. The rule is that if the test statistic is between the two critical values, then we do not reject the null. If the test statistic is not between the two critical values, then we reject H0.\r<BR>\n" );
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document.write( "------------------------------------------------------\r<BR>\n" );
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document.write( "Treatment 1: 14 patients show improvement <BR>\n" );
document.write( "so x1 = 14<BR>\n" );
document.write( "This is out of 22+14+14 = 50 total<BR>\n" );
document.write( "14/50 = 0.28 = 28% of patients show improvement for treatment 1<BR>\n" );
document.write( "phat1 = 0.28\r<BR>\n" );
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document.write( "Treatment 2: 21 patients show improvement <BR>\n" );
document.write( "so x2 = 21<BR>\n" );
document.write( "This is out of 14+15+21 = 50 total<BR>\n" );
document.write( "21/50 = 0.42 = 42% of patients show improvement for treatment 2<BR>\n" );
document.write( "phat2 = 0.42\r<BR>\n" );
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document.write( "The sample sizes of each treatment group is 50, so<BR>\n" );
document.write( "n1 = 50<BR>\n" );
document.write( "n2 = 50\r<BR>\n" );
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document.write( "------------------------------------------------------\r<BR>\n" );
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document.write( "Let's find the \"average\" sample proportion value and call this qhat\r<BR>\n" );
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document.write( "qhat = (x1+x2)/(n1+n2)<BR>\n" );
document.write( "qhat = (14+21)/(50+50)<BR>\n" );
document.write( "qhat = 0.35 .... use a calculator here\r<BR>\n" );
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document.write( "Using that value of qhat, we can compute the standard error\r<BR>\n" );
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document.write( "SE = sqrt(qhat*(1-qhat)*(1/n1 + 1/n2))<BR>\n" );
document.write( "SE = sqrt(0.35*(1-0.35)*(1/50 + 1/50))<BR>\n" );
document.write( "SE = 0.0953939201417 .... use a calculator here\r<BR>\n" );
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document.write( "-------------------------------------------------------\r<BR>\n" );
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document.write( "Now onto the test statistic\r<BR>\n" );
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document.write( "z = (phat1-phat2)/(SE)<BR>\n" );
document.write( "z = (0.28 - 0.42)/(0.0953939201417)<BR>\n" );
document.write( "z = -1.4675987714106<BR>\n" );
document.write( "z = -1.47 ... rounding to two decimal places\r<BR>\n" );
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document.write( "The level of significance is 5%. Alpha = 0.05<BR>\n" );
document.write( "confidence level = 1 - alpha<BR>\n" );
document.write( "confidence level = 1 - 0.05<BR>\n" );
document.write( "confidence level = 0.95<BR>\n" );
document.write( "confidence level = 95%\r<BR>\n" );
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document.write( "Use a table like <a  rel=nofollow HREF=\"http://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf\">this one</a> to find the critical values. Look at the bottom of the page. Locate the 95% confidence level. Look directly above it and you'll see the value 1.960\r<BR>\n" );
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document.write( "Since this is a two-tailed test, this means the critical values are -1.960 and 1.960\r<BR>\n" );
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document.write( "Go back to the rule stated above. Since -1.47 is definitely between -1.960 and 1.960, this means we do not reject the null. We must conclude that p1 = p2. There isn't significant evidence to prove it wrong. \r<BR>\n" );
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document.write( "So the final answer is <BR>\n" );
document.write( "<font color=red>B. There is Not significant evidence, alpha 0.05 to show that there is a difference in the proportion of patients who show improvements between treatments 1 and 2.</font>\r<BR>\n" );
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document.write( "Further reading:<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://onlinecourses.science.psu.edu/stat414/node/268\">https://onlinecourses.science.psu.edu/stat414/node/268</a>\r<BR>\n" );
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