document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1201385>Question 1201385</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The prime Minister of a small Caribbean Island stated that 95% of the population was vaccinated from the Covid-19 virus. The opposition believes that the Minister is overstating the proportion of vaccinated citizens. He randomly selects 300 citizens and found that 240 of them were fully vaccinated. \r<BR>\n" );
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document.write( "iv. Calculate the value of the test statistics. <BR>\n" );
document.write( "v. At the 5% level of significance, determine if the Politian overstated the proportion of vaccinated citizen. Use the classical approach.<BR>\n" );
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document.write( "Part (iv)\r<BR>\n" );
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document.write( "p = 0.95 = hypothesized population proportion of people who got vaccinated<BR>\n" );
document.write( "phat = sample proportion = 240/300 = 0.80<BR>\n" );
document.write( "n = 300 = sample size\r<BR>\n" );
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document.write( "SE = standard error<BR>\n" );
document.write( "SE = sqrt(p*(1-p)/n)<BR>\n" );
document.write( "SE = sqrt(0.95*(1-0.95)/300)<BR>\n" );
document.write( "SE = 0.01258305739211<BR>\n" );
document.write( "This value is approximate.\r<BR>\n" );
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document.write( "Test statistic:<BR>\n" );
document.write( "z = (phat - p)/SE<BR>\n" );
document.write( "z = (0.80 - 0.95)/0.01258305739211<BR>\n" );
document.write( "z = -11.9207912135929<BR>\n" );
document.write( "z = -11.92\r<BR>\n" );
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document.write( "<font color=red>Test statistic: z = -11.92 approximately</font>\r<BR>\n" );
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document.write( "Part (v)\r<BR>\n" );
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document.write( "p = population proportion of people who got vaccinated\r<BR>\n" );
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document.write( "Null hypothesis: p = 0.95<BR>\n" );
document.write( "Alternative hypothesis: p < 0.95\r<BR>\n" );
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document.write( "The prime minister's claim is in the null hypothesis.<BR>\n" );
document.write( "The opposition's claim is in the alternative hypothesis.\r<BR>\n" );
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document.write( "This is because the opposition believes the 95% vaccination rate is overstated (i.e. the value of p is lower).\r<BR>\n" );
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document.write( "This is a left-tailed test due to the \"less than\" sign in the alternative hypothesis.<BR>\n" );
document.write( "If the test statistic is to the left of the critical value, then we reject the null.\r<BR>\n" );
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document.write( "At the 5% level of significance, the left-tailed critical value is approximately z = -1.645 (use a table or stats calculator to determine this)<BR>\n" );
document.write( "P(Z < -1.645) = 0.05 approximately<BR>\n" );
document.write( "5% of the area under the standard normal Z curve is to the left of z = -1.645\r<BR>\n" );
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document.write( "We found that<BR>\n" );
document.write( "test statistic = -11.92<BR>\n" );
document.write( "critical value = -1.645 <BR>\n" );
document.write( "both of which are approximate\r<BR>\n" );
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document.write( "The test statistic is to the left of the critical value.<BR>\n" );
document.write( "<font color=red>We're in the rejection region.</font><BR>\n" );
document.write( "Therefore we <font color=red>reject the null </font> and conclude p < 0.95 is the case.\r<BR>\n" );
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document.write( "<font color=red>Conclusion: it appears the prime minister has likely overstated the vaccination rate. It's likely less than 95%.</font><BR>\n" );
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