Question

Express the confidence interval 0.38

Answer 1

The correct option is B.

The confidence interval expressed in the form of
`\(\hat{p} \pm E\)` is
`\(0.46 \pm 0.08\)`.

To express the confidence interval in the form of
`\(\hat{p} \pm E\)`, where
`\(\hat{p}\)` is the point estimate of the population proportion and
`\(E\)` is the margin of error, you would follow these steps:

A. Calculate the point estimate,
`\(\hat{p}\)`, which is the midpoint of the confidence interval.

B.Calculate the margin of error,
`\(E\)`, which is the difference between the point estimate and either end of the confidence interval.

Given the confidence interval
`\(0.38 < p < 0.54\)`, here's how you would calculate it:

A. Find the midpoint
`(\(\hat{p}\))`:

• The midpoint is
  `\((0.38 + 0.54) / 2\)`.

Certainly, let's go through the steps to calculate the midpoint,
`\(\hat{p}\)`, which is the point estimate of the population proportion:

1. Identify the lower and upper bounds of the confidence interval:

• The lower bound is given as
  `\(0.38\)`.
• The upper bound is given as
  `\(0.54\)`.

2. Calculate the midpoint
`(\(\hat{p}\))`:

• To find the midpoint, you add the lower and upper bounds together and then divide by
  `\(2\)`.
• The formula to calculate the midpoint is
  `\(\hat{p} = \frac{{\text{lower bound} + \text{upper bound}}}{2}\)`.

3. Apply the values to the formula:

• Substitute the given values into the formula:
  `\(\hat{p} = \frac{{0.38 + 0.54}}{2}\)`.

4. Calculate the midpoint
`(\(\hat{p}\))`:

• Perform the addition and division:
  `\(\hat{p} = \frac{{0.38 + 0.54}}{2} = \frac{{0.92}}{2}\)`.

5. Find the value of
`\(\hat{p}\)`:

• Divide
  `\(0.92\)` by
  `\(2\)` to get the midpoint.

The midpoint calculation step by step is:

`\[\hat{p} = \frac{{0.38 + 0.54}}{2} = \frac{{0.92}}{2} = 0.46\]`

Thus, the midpoint of the confidence interval, which is the point estimate
`\(\hat{p}\)`, is
`\(0.46\)`.

B. Find the margin of error
`(\(E\))`:

• The margin of error is
  `\(\hat{p} - 0.38\)` or
  `\(0.54 - \hat{p}\)` (both should give you the same result since
  `\(\hat{p}\)` is in the middle).

To calculate the margin of error
`\(E\)`, you can use either the lower or upper bound of the confidence interval and the point estimate
`\(\hat{p}\)`. Since the point estimate is the midpoint of the interval, the margin of error will be the same whether you subtract the lower bound from
`\(\hat{p}\)` or subtract
`\(\hat{p}\)` from the upper bound. Here's how you do it step by step:

1. Identify the point estimate
`(\(\hat{p}\))`:

• From the previous step, we found that
  `\(\hat{p} = 0.46\)`.

2. Calculate the margin of error using the lower bound:

• The formula to calculate the margin of error using the lower bound is
  `\(E = \hat{p} - \text{lower bound}\)`.

• Substitute the given values into the formula:
  `\(E = 0.46 - 0.38\)`.

3. Alternatively, calculate the margin of error using the upper bound:

• The formula to calculate the margin of error using the upper bound is
  `\(E = \text{upper bound} - \hat{p}\)`.
• Substitute the given values into the formula:
  `\(E = 0.54 - 0.46\)`.

4. Perform the subtraction to find
`\(E\)`:

• Using the lower bound:
  `\(E = 0.46 - 0.38\)`.
• Using the upper bound:
  `\(E = 0.54 - 0.46\)`.

5. Identify the value of
`\(E\)`:

• Both calculations should yield the same result for
  `\(E\)`.

Let's perform both calculations to confirm that they yield the same margin of error:

The margin of error
`\(E\)` calculated using both the lower and upper bounds is
`\(0.08\)`.

This confirms that the margin of error is the same whether you use the lower or upper bound because the point estimate
`\(\hat{p}\)` is the exact midpoint of the interval:

• Using the lower bound:
  `\(E = 0.46 - 0.38 = 0.08\)`
• Using the upper bound:
  `\(E = 0.54 - 0.46 = 0.08\)`

Hence, the margin of error
`\(E\)` is
`\(0.08\)`.

Let's calculate these values.

The point estimate
`\(\hat{p}\)` of the population proportion is
`\(0.46\)` and the margin of error
`\(E\)` is
`\(0.08\)`.

Therefore, The answer is
`$0.46 \pm 0.08$`.

The complete question is here:

Express the confidence interval
`$0.38 < p < 0.54$` in the form of
`$\hat{p} \pm E$`.

A.
`$0.38 \pm 0.16$`

B.
`$0.46 \pm 0.08$`

C.
`$0.38 \pm 0.08$`

D.
`$0.46 \pm 0.016$`

Answer 2

The confidence interval 0.38 < p < 0.54 can be expressed as:

(p) hat ± E = 0.46 ± 0.08

How to express the confidence interval in required format

To express the confidence interval 0.38 < p < 0.54 in the form of (p) hat ± E, find the point estimate (p) hat and the margin of error (E).

The point estimate (p) hat is the average of the upper and lower bounds of the confidence interval.

In this case, (p) hat can be calculated as:

(p) hat = (0.38 + 0.54) / 2 = 0.46

The margin of error (E) is half the width of the confidence interval. It can be calculated by subtracting the lower bound value from the point estimate:

E = (0.54 - 0.38) / 2 = 0.08

Therefore, the confidence interval 0.38 < p < 0.54 can be expressed as:

(p) hat ± E = 0.46 ± 0.08

Express the confidence interval using the indicated format. Express the confidence interval 0.38 < p < 0.54 in the form of (p) hat ± E.