Question

question 5
1 pts
a total of 10,000 professionals attend a medical conference, and a random sample of 120 of these conference attendees are surveyed. when asked if they were satisfied with the chosen conference venue, 75% of those who were surveyed said \yes.\ which one of the following statements will be correct if a 90% confidence interval is constructed based on this data?
because the margin of error is 0.065, this means that if we add and subtract 6.5 from 0.75, well obtain an interval that has a good chance of including the true population proportion.
we can be 90% confident the interval from 0.685 to 0.815 includes the true sample proportion.
increasing the size of the sample will reduce the width of the interval and probably reduce the center of the interval as well.
because n = 10 000, the margin of error will be approximately 0.010.
none of the above answer options are correct.

Explanation:

• First option: Margin of error is added and subtracted from the sample proportion (not the raw number). Here, \(0.75\pm0.065\) (not \(0.75\pm6.5\)) gives the confidence interval for the population proportion. So this option is wrong.
• Second option: Confidence intervals are constructed to estimate the population proportion, not the sample proportion. The sample proportion is known (\(p = 0.75\)). So this option is wrong.
• Third option: Increasing the sample size reduces the width of the interval (because \(E=z\sqrt{\frac{p(1 - p)}{n}}\), as \(n\) increases, \(E\) decreases). But the center of the interval is the sample proportion \(\hat{p}\), which is not affected by the sample size (as long as the sampling is random). So this option is wrong.
• Fourth option: The formula for margin of error for a proportion is \(E = z\sqrt{\frac{p(1 - p)}{n}}\). Here, \(n = 120\) (the sample size, not the population size \(N=10000\)). For a \(90\%\) confidence interval, \(z=1.645\), \(p = 0.75\), \(n = 120\). \(E=1.645\sqrt{\frac{0.75\times(1 - 0.75)}{120}}\approx1.645\times0.0395\approx0.065

eq0.010\). So this option is wrong.

Answer:

None of the above answer options are correct.