Use the sample data and confidence level given below to complete parts (a) through (d). A research institute poll asked respondents if they felt vulnerable to identity theft. In the poll, n equals 1036 and x equals 583 who said "yes." Use a 90 % confidence level.

Required:
a. Find the best point estimate of the population proportion p.
b. Identify the value of the margin of error E =_______
c. Construct the confidence interval.
d. Write a statement that correctly interprets the confidence interval.

1. One has 99% confidence that the sample proportion is equal to the population proportion.
2. There is a 99% chance that the true value of the population proportion will fall between the lower bound and the upper bound.
3. One has 99% confidence that the interval from the lower bound to the upper bound actually does contain the true value of the population proportion.

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mtosi17 mtosi17 · Answer 1 · 2020-07-06T03:56:11+02:00

Answer:

a. p=0.562

b. E = 0.0253

c. The 90% confidence interval for the population proportion is (0.537, 0.587).

d. We have 90% confidence that the interval (0.537, 0.587) contains the true value of the population proportion.

Step-by-step explanation:

We have to calculate a 90% confidence interval for the proportion.

The sample proportion is p=0.562.

[tex]p=X/n=583/1038=0.562[/tex]

The standard error of the proportion is:

[tex]\sigma_p=\sqrt{\dfrac{p(1-p)}{n}}=\sqrt{\dfrac{0.562*0.438}{1038}}\\\\\\ \sigma_p=\sqrt{0.000237}=0.0154[/tex]

The critical z-value for a 90% confidence interval is z=1.645.

The margin of error (MOE) can be calculated as:

[tex]MOE=z\cdot \sigma_p=1.645 \cdot 0.0154=0.0253[/tex]

Then, the lower and upper bounds of the confidence interval are:

[tex]LL=p-z \cdot \sigma_p = 0.562-0.0253=0.537\\\\UL=p+z \cdot \sigma_p = 0.562+0.0253=0.587[/tex]

The 90% confidence interval for the population proportion is (0.537, 0.587).

We have 90% confidence that the interval contains the true value of the population proportion.