Question 1201385: 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.
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.
Answer by math_tutor2020(3817)   (Show Source):
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Part (iv)
p = 0.95 = hypothesized population proportion of people who got vaccinated
phat = sample proportion = 240/300 = 0.80
n = 300 = sample size
SE = standard error
SE = sqrt(p*(1-p)/n)
SE = sqrt(0.95*(1-0.95)/300)
SE = 0.01258305739211
This value is approximate.
Test statistic:
z = (phat - p)/SE
z = (0.80 - 0.95)/0.01258305739211
z = -11.9207912135929
z = -11.92
Test statistic: z = -11.92 approximately
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Part (v)
p = population proportion of people who got vaccinated
Null hypothesis: p = 0.95
Alternative hypothesis: p < 0.95
The prime minister's claim is in the null hypothesis.
The opposition's claim is in the alternative hypothesis.
This is because the opposition believes the 95% vaccination rate is overstated (i.e. the value of p is lower).
This is a left-tailed test due to the "less than" sign in the alternative hypothesis.
If the test statistic is to the left of the critical value, then we reject the null.
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)
P(Z < -1.645) = 0.05 approximately
5% of the area under the standard normal Z curve is to the left of z = -1.645
We found that
test statistic = -11.92
critical value = -1.645
both of which are approximate
The test statistic is to the left of the critical value.
We're in the rejection region.
Therefore we reject the null and conclude p < 0.95 is the case.
Conclusion: it appears the prime minister has likely overstated the vaccination rate. It's likely less than 95%.
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