Question

(b) find the 95% confidence interval of the mean of these pounds. round intermediate answers to at least three decimal places. round your final answers to one decimal place.
30.9 < μ < 33.1
part: 2 / 4
part 3 of 4
(c) find the 99% confidence interval of the mean of these pounds. round intermediate answers to at least three decimal places. round your final answers to one decimal place.
< μ <

Explanation:

To solve for the 99% confidence interval, we first need to know the sample mean (\(\bar{x}\)) and the standard error (or the sample standard deviation and sample size). From part (b), we can infer that the sample mean \(\bar{x}\) is the midpoint of the 95% confidence interval.

Step 1: Calculate the sample mean (\(\bar{x}\))

The 95% confidence interval is \(30.9 < \mu < 33.1\). The midpoint (sample mean) is:

$$ \bar{x} = \frac{30.9 + 33.1}{2} = \frac{64}{2} = 32.0 $$

Step 2: Determine the margin of error for 95% confidence interval

The margin of error (\(E_{95}\)) for the 95% confidence interval is:

$$ E_{95} = 33.1 - 32.0 = 1.1 $$

For a 95% confidence interval, the critical value \(z_{95}\) (assuming large sample or normal distribution) is approximately 1.96. The standard error (SE) is related to the margin of error by \(E = z \times SE\). So,

$$ SE = \frac{E_{95}}{z_{95}} = \frac{1.1}{1.96} \approx 0.561 $$

Step 3: Find the critical value for 99% confidence interval

For a 99% confidence interval, the critical value \(z_{99}\) (for a normal distribution) is approximately 2.576.

Step 4: Calculate the margin of error for 99% confidence interval

Using the standard error from Step 2, the margin of error \(E_{99}\) for 99% confidence interval is:

$$ E_{99} = z_{99} \times SE = 2.576 \times 0.561 \approx 1.445 $$

Step 5: Calculate the 99% confidence interval

The 99% confidence interval is:

$$ \bar{x} - E_{99} < \mu < \bar{x} + E_{99} $$

Substituting the values:

$$ 32.0 - 1.445 < \mu < 32.0 + 1.445 $$

$$ 30.555 < \mu < 33.445 $$

Rounding to one decimal place:

$$ 30.6 < \mu < 33.4 $$

Answer:

\(30.6 < \mu < 33.4\)