Our homework is about creating a four-step model hypothesis. The problem is,
Test the hypothesis that there is an increase in the percentage of Koreans who are willing to adopt a dog. Conduct a survey among 30 people regarding their answers on the matter. (I conducted the survey and the results = 30 out 30 are willing)
It is found on average that 39% of Koreans are willing to adopt a dog with a population standard deviation of 48.
Use 5% level of significance.
My solution: enter image description here
If I will use the formula, the interpretation would be Fail to reject Ho = there is insufficient evidence to conclude that there is an increase in the percentage of Koreans who are willing to adopt a dog. But this doesn't make sense because I got a 100% from my survey...
the first typo in the exercise is that standard deviation is $0.48$ or 48%, as you prefer (better it is if you set it as 0.49). It cannot be 48 because the pupulation is bernulli...thus st dev is $\sqrt{p(1-p)}=\sqrt{0.39(1-0.39)}\approx0.4877$
the t-stat is, as usual
$$\frac{1-0.39}{0.4877}\sqrt{30}=6.85$$
which is highly significative...no calculations or tables are needed when the t-stat is greater than 3...you are extremely in the right tail
Concluding: the increase is significative at 5%, but at 1% too and more...it is significative also at a $3.6\times10^{-10}\%$ level too
In order to be allowed to use this "gaussian" test-statistic you have previously to check that $np$ and $n(1-p)$ are $\ge 5$ and you are good because
$$np=30\times0.39=11.7$$
and
$$n(1-p)=30\times0.61=18.3$$