Questions: Here are summary statistics for randomly selected weights of newborn girls: n=36, x̄=3150.0 g, s=695.5 g. Use a confidence level of 95% to complete parts (a) through (d) below. b. Find the margin of error. E=227.2 g (Round to one decimal place as needed.) c. Find the confidence interval estimate of μ. 2922.8 g<μ<3377.2 g (Round to one decimal place as needed.) d. Write a brief statement that interprets the confidence interval. Choose the correct answer below. A. One has 95% confidence that the sample mean weight of newborn girls is equal to the population mean weight of newborn girls. B. Approximately 95% of sample mean weights of newborn girls will fall between the lower bound and the upper bound. C. One has 95% confidence that the interval from the lower bound to the upper bound contains the true value of the population mean weight of newborn girls. D. There is a 95% chance that the true value of the population mean weight of newborn girls will fall between the lower bound and the upper bound.

Here are summary statistics for randomly selected weights of newborn girls: n=36, x̄=3150.0 g, s=695.5 g. Use a confidence level of 95% to complete parts (a) through (d) below. b. Find the margin of error. E=227.2 g (Round to one decimal place as needed.) c. Find the confidence interval estimate of μ. 2922.8 g<μ<3377.2 g (Round to one decimal place as needed.) d. Write a brief statement that interprets the confidence interval. Choose the correct answer below. A. One has 95% confidence that the sample mean weight of newborn girls is equal to the population mean weight of newborn girls. B. Approximately 95% of sample mean weights of newborn girls will fall between the lower bound and the upper bound. C. One has 95% confidence that the interval from the lower bound to the upper bound contains the true value of the population mean weight of newborn girls. D. There is a 95% chance that the true value of the population mean weight of newborn girls will fall between the lower bound and the upper bound.
Transcript text: Here are summary statistics for randomly selected weights of newborn girls: $\mathrm{n}=36, \bar{x}=3150.0 \mathrm{~g}, \mathrm{~s}=695.5 \mathrm{~g}$. Use a confidence level of $95 \%$ to complete parts (a) through ( $\boldsymbol{d}$ ) below. b. Find the margin of error. \[ \mathrm{E}=227.2 \mathrm{~g} \] (Round to one decimal place as needed.) c. Find the confidence interval estimate of $\mu$. \[ 2922.8 \mathrm{~g}<\mu<3377.2 \mathrm{~g} \] (Round to one decimal place as needed.) d. Write a brief statement that interprets the confidence interval. Choose the correct answer below. A. One has $95 \%$ confidence that the sample mean weight of newborn girls is equal to the population mean weight of newborn girls. B. Approximately $95 \%$ of sample mean weights of newborn girls will fall between the lower bound and the upper bound. C. One has $95 \%$ confidence that the interval from the lower bound to the upper bound contains the true value of the population mean weight of newborn girls. D. There is a $95 \%$ chance that the true value of the population mean weight of newborn girls will fall between the lower bound and the upper bound.
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Solution

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Solution Steps

Step 1: Calculate the Margin of Error

To find the margin of error \( E \), we use the formula:

\[ E = Z \times \frac{s}{\sqrt{n}} \]

Given that the Z-score for a 95% confidence level is \( Z = 2.0 \), the sample standard deviation \( s = 695.5 \, \text{g} \), and the sample size \( n = 36 \):

\[ E = 2.0 \times \frac{695.5}{\sqrt{36}} = 227.2 \, \text{g} \]

Thus, the margin of error is:

\[ \boxed{E = 227.2 \, \text{g}} \]

Step 2: Calculate the Confidence Interval

The confidence interval for the population mean \( \mu \) is calculated using the formula:

\[ \bar{x} \pm Z \times \frac{s}{\sqrt{n}} \]

Substituting the values:

\[ 3150.0 \pm 2.0 \times \frac{695.5}{\sqrt{36}} = 3150.0 \pm 227.2 \]

Calculating the lower and upper bounds:

\[ \text{Lower Bound} = 3150.0 - 227.2 = 2922.8 \, \text{g} \] \[ \text{Upper Bound} = 3150.0 + 227.2 = 3377.2 \, \text{g} \]

Thus, the confidence interval is:

\[ \boxed{2922.8 \, \text{g} < \mu < 3377.2 \, \text{g}} \]

Step 3: Interpret the Confidence Interval

The correct interpretation of the confidence interval is:

C. One has \( 95\% \) confidence that the interval from the lower bound to the upper bound contains the true value of the population mean weight of newborn girls.

Final Answer

  • Margin of Error: \( \boxed{E = 227.2 \, \text{g}} \)
  • Confidence Interval: \( \boxed{2922.8 \, \text{g} < \mu < 3377.2 \, \text{g}} \)
  • Interpretation: The answer is C.
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