Question

Several years​ ago, 36​% of parents who had children in grades​ K-12 were satisfied with the quality of education the students receive. A recent poll asked 1 comma 205 parents who have children in grades​ K-12 if they were satisfied with the quality of education the students receive. Of the 1 comma 205 ​surveyed, 498 indicated that they were satisfied. Construct a 95​% confidence interval to assess whether this represents evidence that​ parents' attitudes toward the quality of education have changed.

What are the null and alternative​ hypotheses?

Upper H 0​: p ?

Upper H 1​: p ?

​(Round to two decimal places as​ needed.)

Use technology to find the 99​% confidence interval.

The lower bound is

The upper bound is

​(Round to two decimal places as​ needed.)

What is the correct​ conclusion?

A.

Since the interval does not contain the proportion stated in the null​ hypothesis, there is sufficient evidence that​ parents' attitudes toward the quality of education have changed.

B.

Since the interval contains the proportion stated in the null​ hypothesis, there is sufficient evidence that​ parents' attitudes toward the quality of education have changed.

C.

Since the interval contains the proportion stated in the null​ hypothesis, there is insufficient evidence that​ parents' attitudes toward the quality of education have changed.

D.

Since the interval does not contain the proportion stated in the null​ hypothesis, there is insufficient evidence that​ parents' attitudes toward the quality of education have changed.

Answer 1

Below are the null and alternative Hypothesis,
Null Hypothesis, H0: p = 0.36
Alternative Hypothesis, Ha: p ≠ 0.36

sample proportion, = 0.4133
sample size, n = 1205
Standard error, SE = sqrt(pcap * (1 - pcap)/n)
SE = sqrt(0.4133 * (1 - 0.4133)/1205) = 0.0142

Given CI level is 99%, hence α = 1 - 0.99 = 0.01
α/2 = 0.01/2 = 0.005, Zc = Z(α/2) = 2.576

Margin of Error, ME = zc * SE
ME = 2.576 * 0.0142
ME = 0.0366

CI = (pcap - z*SE, pcap + z*SE)
CI = (0.4133 - 2.576 * 0.0142 , 0.4133 + 2.576 * 0.0142)
CI = (0.3767 , 0.4499)

Lower bound = 0.38
Upper bound = 0.45

A.

Since the interval does not contain the proportion stated in the null​ hypothesis, there is sufficient evidence that​ parents' attitudes toward the quality of education have changed.