document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1201383>Question 1201383</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( "i. Calculate a 99% confidence interval for the true proportion of all citizens who were vaccinated.<BR>\n" );
document.write( "ii. Interpret you answer in i).<BR>\n" );
document.write( "iii. State the null and alternative hypothesis of this test.  <BR>\n" );
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document.write( "Part (i)\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( "n = sample size = 300<BR>\n" );
document.write( "phat = sample proportion of people who got vaccinated = 240/300 = 0.80\r<BR>\n" );
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document.write( "At 99% confidence, the z critical value is roughly z = 2.576 <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 that value. Look at the bottom row labeled \"Z\" and above the 99% confidence level.<BR>\n" );
document.write( "A stats calculator can also compute this value.\r<BR>\n" );
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document.write( "Compute the margin of error for the proportion.<BR>\n" );
document.write( "E = z*sqrt(phat*(1-phat)/n)<BR>\n" );
document.write( "E = 2.576*sqrt(0.80*(1-0.80)/300)<BR>\n" );
document.write( "E = 0.0594901717373<BR>\n" );
document.write( "E = 0.059490<BR>\n" );
document.write( "This value is approximate.\r<BR>\n" );
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document.write( "Now we can compute the boundaries.<BR>\n" );
document.write( "L = lower boundary of the confidence interval<BR>\n" );
document.write( "L = phat - E<BR>\n" );
document.write( "L = 0.80 - 0.059490<BR>\n" );
document.write( "L = 0.74051<BR>\n" );
document.write( "and<BR>\n" );
document.write( "U = upper boundary of the confidence interval<BR>\n" );
document.write( "U = phat + E<BR>\n" );
document.write( "U = 0.80 + 0.059490<BR>\n" );
document.write( "U = 0.85949<BR>\n" );
document.write( "These values are approximate.\r<BR>\n" );
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document.write( "The 99% confidence interval in the format (L, U) is approximately <font color=red>(0.74051, 0.85949)</font>\r<BR>\n" );
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document.write( "The 99% confidence interval in the format L < p < U is approximately <font color=red>0.74051 < p < 0.85949</font><BR>\n" );
document.write( "This second format is a bit more descriptive in terms of which population parameter we're trying to measure. \r<BR>\n" );
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document.write( "Side note:<BR>\n" );
document.write( "An alternative confidence interval format is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=phat+%2B-+E ALIGN=MIDDLE  ALT=\"phat+%2B-+E\"   DISABLED_style= \"max-width:90%; height:auto;\" > which in this case is roughly <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.80+%2B-+0.059490 ALIGN=MIDDLE  ALT=\"0.80+%2B-+0.059490\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Part (ii)\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( "In the previous section we found 0.74051 < p < 0.85949\r<BR>\n" );
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document.write( "We are <font color=red>99% confident</font> the population proportion <font color=red>p is somewhere between 0.74051 and 0.85949</font>\r<BR>\n" );
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document.write( "Meaning we are 99% confident the true percentage of people who got vaccinated is somewhere between 74.051% and 85.949%<BR>\n" );
document.write( "Each percentage is approximate.\r<BR>\n" );
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document.write( "The percentage 95% is not in the interval between 74.051% and 85.949%, so it appears the opposition is correct in stating the true vaccination rate is below 95%.\r<BR>\n" );
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document.write( "Part (iii)\r<BR>\n" );
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document.write( "p = population proportion of people who got vaccinated<BR>\n" );
document.write( "<font color=red>Null: p = 0.95<BR>\n" );
document.write( "Alternative: p < 0.95</font>\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.<BR>\n" );
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