91影视

Based on Data set 4 鈥淏irth鈥� in Appendix B,the summary statistics for randomly selected weights of newborn girls as,\({\rm{n}}\,{\rm{ = }}\,195\),\(\bar x\, = 32.7\,{\rm{hg}}\),\({\rm{s}}\, = 6.6\,{\rm{hg}}\)

The confidence level is 95%.

A confidence interval is an estimate of interval that may contain true value of a population parameter. It is also known as interval estimate.

The general formula for confidence interval estimate of mean is,

\({\rm{Confidence}}\,\,{\rm{Interval}} = \left( {\bar x - {\rm{E}},\bar x + {\rm{E}}} \right)\,\,\,\,\,\,\,\,...\left( 1 \right)\)\(\)

Where, E is the margin of error, which is calculated as,

\({\rm{E}} = {t_{\frac{\alpha }{2}}} \times \frac{{\rm{s}}}{{\sqrt {\rm{n}} }}\)

If\(\sigma \)is not known and\({\rm{n}} > 30\)then t-distribution is suitable to find the confidence interval.

In this case,\(\sigma \)is unknown and \({\rm{n}} = 195\) which means n>30.So, thet-distribution applies here.

To find the critical value\({t_{\frac{\alpha }{2}}}\), it requires a value for the degrees of freedom.

The degree of freedom is,

\(\)

\(\begin{array}{c}{\rm{degree}}\,\,\,{\rm{of}}\,\,\,{\rm{freedom}}\,\,\, = n - 1\\ = 195 - 1\\ = 194\end{array}\)

The 95% confidence level corresponds to\(\alpha = 0.05\), so there is an area of 0.025 in each of the two tails of the t-distribution.

Referring to Table A-3 critical value of t-distribution, the critical value of\({{\bf{t}}_{\frac{{\bf{\alpha }}}{{\bf{2}}}}}{\bf{ = }}{{\bf{t}}_{{\bf{0}}{\bf{.025}}}}\)is obtained from the intersection of column with 0.05 for the 鈥淎rea in Two Tails鈥� (or use the same column with 0.025 for the 鈥淎rea in One Tail鈥�)and the row value number of degrees of freedom is 194 , which is 1.972.

The margin of error is calculated as,

\(\begin{array}{c}{\rm{E}} = {t_{\frac{\alpha }{2}}} \times \frac{{\rm{s}}}{{\sqrt {\rm{n}} }}\\ = 1.972 \times \frac{{6.6}}{{\sqrt {195} }}\\ = 0.9322\end{array}\)

The confidence interval is obtained by substituting the value of margin of error in equation (1) as,

\(\begin{array}{c}{\rm{Confidence}}\,\,{\rm{Interval}} = \left( {\bar x - {\rm{E}},\bar x + {\rm{E}}} \right)\\ = \left( {32.7 - 0.9320\,\,,\,\,32.7 + 0.9320} \right)\\ = \left( {31.768\,\,,\,\,33.632} \right)\end{array}\)

Thus, the 95% confidence interval for estimate mean is \(31.8\,{\rm{hg}} < \mu < 33.6\,{\rm{hg}}\).

The confidence interval found in the Exercise-9 with 205 sample values for mean birth weights of girls is\(29.4\,{\rm{hg}} < \mu < 31.4\,{\rm{hg}}\).

The confidence interval for estimate mean with 195 sample values for mean birth weights of boys is\(31.8\,{\rm{hg}} < \mu < 33.6\,{\rm{hg}}\).

So, results in these two cases are almost same.

Also, it does not appear that boys and girls have very different birth weights.