Using the information given, it is found that:
a) 0.39
b) z = 1.96
c) s = 0.0218
d) M = 0.0428
e)
- The lower bound is 0.3472.
- The upper bound is 0.4328.
f) The 95% confidence interval is (0.3472, 0.4328), which means that we are 95% sure that the true proportion of boats who are in violation of one or more safety regulations is (0.3472, 0.4328).
Item a:
The point estimate is the sample proportion, hence:
[tex]\pi = \frac{195}{500} = 0.39[/tex]
Item b:
95% confidence level, hence [tex]\alpha = 0.95[/tex], the critical value z is the value of Z that has a p-value of [tex]\frac{1+0.95}{2} = 0.975[/tex], so [tex]z = 1.96[/tex].
Item c:
The standard error is:
[tex]s = \sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
For this problem, [tex]\pi = 0.39, n = 500[/tex], hence:
[tex]s = \sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
[tex]s = \sqrt{\frac{0.39(0.61)}{500}}[/tex]
[tex]s = 0.0218[/tex]
Item d:
The margin of error is:
[tex]M = zs[/tex]
In this problem, [tex]z = 1.96, s = 0.0218[/tex], hence:
[tex]M = zs = 1.96(0.0218) = 0.0428[/tex]
Item e:
The lower bound is:
[tex]\pi - M = 0.39 - 0.0428 = 0.3472[/tex]
The upper bound is:
[tex]\pi + M = 0.39 + 0.0428 = 0.4328[/tex]
The lower bound is 0.3472.
The upper bound is 0.4328.
Item f:
The 95% confidence interval is (0.3472, 0.4328), which means that we are 95% sure that the true proportion of boats who are in violation of one or more safety regulations is (0.3472, 0.4328).
A similar problem is given at https://brainly.com/question/15136737
