SOLUTION: 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 th
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Question 1201383: 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.
You can put this solution on YOUR website! population is assumed to be 95% vaccinated equals a population ratio of .95.
sample of 300 finds that 240 of them were fully vaccinated.
sample has a ratio of 240 / 300 = .8
standard error for the test is sqrt(.95 * .05 / 300) = .0125830574.
the population ratio is used in the calculation of the standard error for the test.
use the z-score formula to find the probability of getting a score less than .8.
z = (x - m) / s
z is the z-score
x is the sample ratio
m is the population ratio
s is the standard error.
you get z = (.8 - .9) / .0125830574 = -7.947194142.
the 99% confidence interval for the z-score would be plus or minus z = 2.575829303.
since the absolute value of the test z-score is greater than tha absolute value of the critical z-score, the test is considered significant and the conclusion is that ratio is not .95, but something less than that.
p = population proportion of people who got vaccinated
n = sample size = 300
phat = sample proportion of people who got vaccinated = 240/300 = 0.80
At 99% confidence, the z critical value is roughly z = 2.576
Use a table like this https://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf
to get that value. Look at the bottom row labeled "Z" and above the 99% confidence level.
A stats calculator can also compute this value.
Compute the margin of error for the proportion.
E = z*sqrt(phat*(1-phat)/n)
E = 2.576*sqrt(0.80*(1-0.80)/300)
E = 0.0594901717373
E = 0.059490
This value is approximate.
Now we can compute the boundaries.
L = lower boundary of the confidence interval
L = phat - E
L = 0.80 - 0.059490
L = 0.74051
and
U = upper boundary of the confidence interval
U = phat + E
U = 0.80 + 0.059490
U = 0.85949
These values are approximate.
The 99% confidence interval in the format (L, U) is approximately (0.74051, 0.85949)
The 99% confidence interval in the format L < p < U is approximately 0.74051 < p < 0.85949
This second format is a bit more descriptive in terms of which population parameter we're trying to measure.
Side note:
An alternative confidence interval format is which in this case is roughly
p = population proportion of people who got vaccinated
In the previous section we found 0.74051 < p < 0.85949
We are 99% confident the population proportion p is somewhere between 0.74051 and 0.85949
Meaning we are 99% confident the true percentage of people who got vaccinated is somewhere between 74.051% and 85.949%
Each percentage is approximate.
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%.
p = population proportion of people who got vaccinated
Null: p = 0.95
Alternative: 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.
(