Here are summary stastistics for randomly selected weights of newborn girls: n=224,overline{x}=28.3 text{hg},s=7.1 text{hg}. Construct a confidence interval estimate of mean. Use a 98% confidence level. Are these results very different from the confidence interval 26.5 text{hg} < mu < 30.7 text{hg} with only 14 sample values, overline{x}=28.6 hg, and s=2.9 hg? What is the confidence interval for the population mean mu? ?

Question
Confidence intervals
asked 2020-11-07
Here are summary stastistics for randomly selected weights of newborn girls: \(\displaystyle{n}={224},\overline{{{x}}}={28.3}\text{hg},{s}={7.1}\text{hg}\). Construct a confidence interval estimate of mean. Use a 98% confidence level. Are these results very different from the confidence interval \(\displaystyle{26.5}\text{hg}{<}\mu{<}{30.7}\text{hg}\)</span> with only 14 sample values, \(\displaystyle\overline{{{x}}}={28.6}\) hg, and \(\displaystyle{s}={2.9}\) hg? What is the confidence interval for the population mean \(\displaystyle\mu\)? \(\displaystyle?{<}\mu{<}?\)</span> Are the results between the two confidence intervals very different?

Answers (1)

2020-11-08
Step 1 From the provided information, Sample size \(\displaystyle{\left({n}\right)}={244}\) Sample mean \(\displaystyle{\left(\overline{{{x}}}\right)}={28.3}\) hg Sample standard deviation \(\displaystyle{\left({s}\right)}={7.1}\) hg Step 2 Since, the population standard deviation is unknown, therefore, the t distribution will be used. Confidence level = 98% Level of significance \(\displaystyle{\left(\alpha\right)}={1}-{0.98}={0.02}\) The degree of freedom \(\displaystyle={n}–{1}={244}–{1}={243}\) The critical value of t at 243 degree of freedom with 0.01 level of significance from the t value table is 2.34. Step 3 The required 98% confidence interval can be obtained as: \(\displaystyle{C}{I}=\overline{{{x}}}\pm{t}_{{\frac{\alpha}{{2}},{n}-{1}}}{\frac{{{s}}}{{\sqrt{{{n}}}}}}\)
\(\displaystyle={28.3}\pm{\left({2.34}\right)}{\frac{{{7.1}}}{{\sqrt{{{224}}}}}}\)
\(\displaystyle={28.3}\pm{1.1}\)
\(\displaystyle={\left({27.2},{29.4}\right)}\) Thus, the confidence interval is \(\displaystyle{27.2}{<}\mu{<}{29.4}\)</span>. No, the results between the two confidence intervals are not very different. The confidence interval limits are almost similar.
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