Here are summary statistics for randomly selected weights of newborn girls: $$ \mathrm{n}=36, \overline{\mathrm{x}}=3197.2 \mathrm{~g}, \mathrm{~s}=692.6 \mathrm{~g} $$. Use a confidence level of $$ 90 \% $$ to complete parts (a) through (d) below a. Identify the critical value $$ \mathrm{t}_{\alpha / 2} $$ used for finding the margin of error. $$ \mathrm{t}_{\alpha / 2}=1.69 $$ (Round to two decimal places as needed.) b. Find the margin of error. $$ \mathrm{E}=195.0 \mathrm{~g} $$ (Round to one decimal place as needed.) c. Find the confidence interval estimate of $$ \mu $$. $$ \square \mathrm{g}

Question

Here are summary statistics for randomly selected weights of newborn girls: $$ \mathrm{n}=36, \overline{\mathrm{x}}=3197.2 \mathrm{~g}, \mathrm{~s}=692.6 \mathrm{~g} $$. Use a confidence level of $$ 90 \% $$ to complete parts (a) through (d) below a. Identify the critical value $$ \mathrm{t}_{\alpha / 2} $$ used for finding the margin of error. $$ \mathrm{t}_{\alpha / 2}=1.69 $$ (Round to two decimal places as needed.) b. Find the margin of error. $$ \mathrm{E}=195.0 \mathrm{~g} $$ (Round to one decimal place as needed.) c. Find the confidence interval estimate of $$ \mu $$. $$ \square \mathrm{g}<\mu<\square \mathrm{g} $$ (Round to one decimal place as needed.)

UpStudy AI · Accepted Answer

**a. Critical value**  
The critical value is given as  
$$
t_{\alpha/2}=1.69.
$$

**b. Margin of error**  
The margin of error ($$E$$) is computed using  
$$
E=t_{\alpha/2}\left(\frac{s}{\sqrt{n}}\right).
$$  
Substitute the given values:  
$$
E=1.69\left(\frac{692.6}{\sqrt{36}}\right)=1.69\left(\frac{692.6}{6}\right)=1.69(115.4333\dots)\approx195.0\text{ g}.
$$

**c. Confidence interval for $$\mu$$**  
The confidence interval is given by  
$$
\overline{x}-E<\mu<\overline{x}+E.
$$  
Calculate the endpoints:  
$$
\text{Lower bound}=\;3197.2-195.0=3002.2\text{ g},
$$  
$$
\text{Upper bound}=\;3197.2+195.0=3392.2\text{ g}.
$$  
Thus, the $$90\%$$ confidence interval is  
$$
3002.2\text{ g} < \mu < 3392.2\text{ g}.
$$
**a. Critical value** $$ t_{\alpha/2} = 1.69 $$ **b. Margin of error** $$ E = 195.0 \text{ g} $$ **c. Confidence interval for $$\mu$$** $$ 3002.2 \text{ g} < \mu < 3392.2 \text{ g} $$

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