Question

Describe the Confidence Interval 1. In two to three complete sentences, explain what a confidence interval means (in general), as though you were talking to someone who has not taken statistics. 2. In one to two complete sentences, explain what this confidence interval means for this particular study. 3. Construct a confidence interval for each confidence level given. Confidence level EBP/Error Bound Confidence Interval 50% 80% 95% 99% Table 8.7 4. What happens to the EBP as the confidence level increases? Does the width of the confidence interval increase or decrease? Explain why this happens.

Answer 1

1. Confidence Interval: It is an interval which is supposed to contain the true value. If we are talking about 95% confidence interval: If we repeatedly make a 100 sample from the population, then the true value of the parameter will be contained in 95 samples.

2. For this study, this interval will contain the true population proportion of the students who were born in a particular state.

3.

Since for the problem in your exercise, $\hat{p}=\frac{x}{n}=\frac{120}{200}=0.6$

For example the EBP for 95% Confidence level is :$=Z_(1-\alpha /2)*\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

$=1.96*\sqrt{\frac{0.6(1-0.6)}{200}}$

  $=1.96*0.0346$

  $=0.0679$. Likewise, it is calculated for other confidence levels.

Confidence Level	EBP/Error Bound	Confidence Interval lower	Confidence Interval upper	Z-value
50%	0.0234	0.5766	0.6234	0.6745
80%	0.0444	0.5556	0.6444	1.2816
95%	0.0679	0.5321	0.6679	1.9600
99%	0.0892	0.5108	0.6892	2.5758

4. We can see from the table above that when the confidence level increases, the EBP or the bound increases. Accordingly, the width of the interval also increases. This is because EBP is based on the Z value. The value of Z depends on the probability or the confidence level. Since the confidence level is the probability or the area under the curve, the area increases as we increase the confidence level. For example, the Z- value for 50% is 0.6745 and for 99% it is 2.5758 and hence when we multiply the sd with this, the EBP increases.