document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1201407>Question 1201407</A>: <I><SPAN STYLE=\"word-break: break-all;\"> a) 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. i. Calculate a 99% confidence interval for the true proportion of all citizens who were vaccinated. ii. Interpret you answer in i). iii. State the null and alternative hypothesis of this test. iv. Calculate the value of the test statistics. v. At the 5% level of significance, determine if the Politian overstated the proportion of vaccinated citizen. Use the classical approach.  </SPAN>\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" );
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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.\r<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( "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( "Recall the confidence interval we found earlier was<BR>\n" );
document.write( "0.74051 < p < 0.85949<BR>\n" );
document.write( "meaning that the true vaccinated percentage is somewhere between 74.051% and 85.949% (we were 99% confident of this)<BR>\n" );
document.write( "p = 0.95 is nowhere in the interval 0.74051 < p < 0.85949, so it is further evidence that the opposition may be onto something when stating the vaccination rate is below 95%\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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