Question

Days before a presidential​ election, a nationwide random sample of registered voters was taken. Based on this random​ sample, it was reported that​ "52% of registered voters plan on voting for Robert Smith with a margin of error of ±​3%." The margin of error was based on a​ 95% confidence level. Fill in the blanks to obtain a correct interpretation of this confidence interval. We are​ ___________ confident that the​ ___________ of registered voters​ ___________ planning on voting for Robert Smith is between​ ___________ and​ ___________.

Answer 1

The confidence interval means we are 95% certain that the true proportion of voters for Robert Smith lies between 49% and 55%, adjusting for the margin of error of ±3%.

Step-by-step explanation:

We are 95% confident that the proportion of registered voters planning on voting for Robert Smith is between 49% and 55%.

This interpretation means that if we were to take many samples and construct a 95% confidence interval from each sample, we would expect 95% of those intervals to contain the true proportion of voters who favor Robert Smith. The reported margin of error of ±3% gives us the lower and upper bounds of this interval, which we obtain by subtracting and adding the margin of error from the sample proportion, respectively.

Answer 2

We are 95% confident that the percentage of registered voters in the nation planning on voting for Robert Smith is between 49% and 55%.

Step-by-step explanation:

Given that :

Margin of Error = ±3%

Sample Proportion = 52%

Confidence level = 95%

The 95% confidence interval is :

Sample proportion ± margin of error

52% ± 3%

Lower boundary = 52% - 3% = 49%

Upper boundary = 52% + 3% = 55%

The interpretation is that at a given confidence level ; the popukation proportion based on the sample proportion and margin of error is in the confidence interval.