Question 1201383
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Part (i)


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
<a href = "https://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf">https://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf</a>
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 <font color=red>(0.74051, 0.85949)</font>


The 99% confidence interval in the format L < p < U is approximately <font color=red>0.74051 < p < 0.85949</font>
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 {{{phat +- E}}} which in this case is roughly {{{0.80 +- 0.059490}}}


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Part (ii)


p = population proportion of people who got vaccinated


In the previous section we found 0.74051 < p < 0.85949


We are <font color=red>99% confident</font> the population proportion <font color=red>p is somewhere between 0.74051 and 0.85949</font>


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%.


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Part (iii)


p = population proportion of people who got vaccinated
<font color=red>Null: p = 0.95
Alternative: p < 0.95</font>


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.
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