Question 1:
A certain cell phone company believes that the proportion of customers they have among the population in a certain town is 33%. In order to test this, the company run a survey, of which 350 random respondents answered, and 126 respondents, confirmed having this brand.
- At 5% level of significance, is there a strong enough evidence to assume there’s more the actual proportion within the population is greater than 33%?
- At 2% level of significance, is there a strong enough evidence to assume the proportion in the population is different than 33%?
- Find a confidence interval for the true proportion in the population, based upon the results of the survey.
Answer 1:
This is what we know:
- This is the relevant calculation:
And so, the critical Z for 5% level of significance is 1.645, and the result is that 1.645>1.19, so there no good enough evidence to reject the null hypothesis and support the alternative hypothesis: the true proportion isn’t greater than 33% under 5% level of significance.
- Using the same calculation, at 2% level of significance, the relevant critical value would be 2.33, so, in this case as well, there’s no strong enough evidence to reject the null hypothesis and support the alternative hypothesis: at 2% level of significance, the actual proportion isn’t significantly different from 33%.
- That is the calculation for the confidence interval: Since there’s no indication of level of significance, we’ll use 5% as the level of significance.
Question 2:
A politician believes that the level of support in her area is 56% percent.
To validate this, she runs a study of 450 respondents. 272 respondents confirm their support.
- Is there a reasonable reason to believe that there’s a change in the percentage of support of the politician?
- Assuming there’s no prior knowledge of the true level of support of that politician, find a 90% confidence interval for the true proportions of her support.
- What is the minimum sample size required to find a confidence interval with less than 5% difference of her true proportion of support, at 5% level of significance?
Answer 2:
- That’s the relevant P-value for these measures:
The p-value for Z=1.93 is:
So given the result reported, under 5% level of significance, there’s no reasonable reason to conclude there’s a significance change in the true proportion of support.
- With no prior knowledge, the relevant percentage of support we’d assume is 0.50, and so there’s the confidence interval at 90% level of significance:
- There’s the calculation for that:
