Question

Recall that the formula for a confidence interval is Statistc + MOE. Next you will need to interpret the confidence interval. The interpretation of a confidence interval should always include 3 things: the level of confidence, reference to the parameter in context and the calculated interval. Notice that the only difference between this worksheet and the calculating confidence intervals worksheet from module 8 is that you now need to calculate the margin of error. 1. Suppose that a recent study asked a sample of Kentucky residents "Do you believe that evolution should be taught in public schools? Out of the 100 people surveyed (the survey was given to a random sample) 80 responded yes. Confirm that the sample size is large enough, calculate a 95% confidence interval and interpret the interval. (Round your answer to 4 decimal places)

Answer 1

Answer 1:

For sample size to be considered large np and n(1 – p) should be greater than or equal to 10.

In this case n = 100 and p = 80/100 = 0.8

n*p = 100*0.8 = 80 > 10

n(1 – p) = 100*(1-0.8) = 20 > 10

Since both the condition are met, we can say that sample size is large enough.

We need to construct the 95'% confidence interval for the population proportion. We have been provided with the following information about the number of favorable cases: Favorable Cases X 80 Sample Size N- 100 The sample proportion is computed as follows, based on the sample size N of favorable cases X 80: 100 and the number Y 80 N 100 P.8 The critical value for α = 0.05 is:e-aa/ computed as shown below: 2 = 1.06. The corresponding confidence interval is p(1 -p) CI(Proportion) (p-ay- = 0.8(1 0.8) 100 0.8+1.96x 0.S-0.8) 0.8 .. 1.96 x = 100 - (0.722,0.878) Therefore, based on the data provided, the 95% confidence interval for the population proportion is 0.722 < p 0.878, which indicates that we are 95% confident that the true population proportion p is contained by the interval (0.722, 0.878)

Answer 2:

We need to construct the 90% confidence interval for the population proportion. We have been provided with the following information about the number of favorable cases: 80 Favorable Cases X Sample Size N- 100 The sample proportion is computed as follows, based on the sample size N of favorable cases X 80 100 and the number Y 80 PN100.8 The critical value for α-: 0.1 is :c-~ computed as shown below -= 1.645. The corresponding confidence interval is p(l-p) p(1 - p) CI(Proportion) = {p -se 0.8) 0.8 -1.645x 08(1 O.8) 0.8+1.645 x 0.8 100 100 -(0.734,0.866) Therefore, based on the data provided, the 90% confidence interval for the population proportion is 0.734-< p 0.866, which indicates that we are 90% confident that the true population proportion p is contained by the interval (0.734, 0.866)

Answer 3:

We need to construct the 99% confidence interval for the population proportion. We have been provided with the following information about the number of favorable cases: Favorable Cases X Sample Size N- 80 100 The sample proportion is computed as follows, based on the sample size N of favorable cases X 80 100 and the number X 80 PN100.8 The critical value fora = 0.01 is:c-aa/2 = 2.576. The corresponding confidence interval is computed as shown below: CI(Proportion) n 0.8(1 0.8) 100 0.8(1 0.8) 100 0.8 2.576 x 0.82.576 x -(0.697,0.903) Therefore, based on the data provided, the 99% confidence interval for the population proportion is 0.697 < p 0.903, which indicates that we are 99% confident that the true population proportion p is contained by the interval (0.697,0.903)

Answer 4:

90% Confidence Interval is (0.734, 0.866)

95% Confidence Interval is (0.722, 0.878)

99% Confidence Interval is (0.697, 0.903)

From the Answer 1 to 3, we can see that as the level of confidence increases, width of confidence interval also increases. This is because as the level of confidence increases, critical z value increases. The increase in critical z value increase the margin of error.