91影视

Assume that the populations are normally distributed and that the samples are chosen separately. The

Two-sample confidence interval for \({\mu _1} - {\mu _2}\)

With confidence level \(100(1 - \alpha )\% is \)

\(\bar x - \bar y \pm {t_{\alpha /2,\nu }} \times \sqrt {\frac{{s_1^2}}{m} + \frac{{s_2^2}}{n}} \)

\(\nu = \frac{{{{\left( {\frac{{s_1^2}}{m} + \frac{{s_2^2}}{n}} \right)}^2}}}{{\frac{{{{\left( {s_1^2/m} \right)}^2}}}{{m - 1}} + \frac{{{{\left( {s_2^2/n} \right)}^2}}}{{n - 1}}}}\)

The number must be rounded to the nearest integer. An upper/lower bound is obtained by substituting \({z_{\alpha /2}}\) with \({z_\alpha } \) and \( \pm \)with only \( + \) and \( - \).

The degrees of freedom can be calculated using the formula below.

\(\begin{array}{l}\nu = \frac{{{{\left( {\frac{{s_1^2}}{m} + \frac{{s_2^2}}{n}} \right)}^2}}}{{\frac{{{{\left( {s_1^2/m} \right)}^2}}}{{m - 1}} + \frac{{{{\left( {s_2^2/n} \right)}^2}}}{{n - 1}}}} = \frac{{{{\left( {\frac{{5.0{3^2}}}{6} + \frac{{5.3{8^2}}}{6}} \right)}^2}}}{{\frac{{{{\left( {5.0{3^2}/6} \right)}^2}}}{{6 - 1}} + \frac{{{{\left( {5.3{8^2}/6} \right)}^2}}}{{6 - 1}}}}\\ = \frac{{{{(4.2168 + 4.8241)}^2}}}{{\frac{{{{(4.2168)}^2}}}{5} + \frac{{4.824{1^2}}}{5}}} = 9.96\end{array}\)

The number must be rounded down to 9 in order to acquire the matching degrees of freedom, implying that.

\(\nu = 9\)

The only value that is lacking is $t alpha / 2, nu$. The value of \({t_{\alpha /2,\nu }}. For 95\% \) confidence interval is \(\alpha \), \(0.05\) hence \(\alpha /2 = 0.025\). The value is based on the table in the appendix.

\({t_{\alpha /2,v}} = {t_{0.025,9}} = 2.262.\)

Finally, the \(95\) confidence interval can be computed

\(\begin{array}{l}\left( {\bar x - \bar y - {t_{\alpha /2,\nu }} \times \sqrt {\frac{{s_1^2}}{m} + \frac{{s_2^2}}{n}} ,\bar x + \bar y - {t_{\alpha /2,\nu }} \times \sqrt {\frac{{s_1^2}}{m} + \frac{{s_2^2}}{n}} } \right)\\ = \left( {115.7 - 129.3 - 2.262 \times \sqrt {\frac{{5.0{3^2}}}{6} + \frac{{5.3{8^2}}}{6}} ,115.7 - 129.3 + 2.262 \times \sqrt {\frac{{5.0{3^2}}}{6} + \frac{{5.3{8^2}}}{6}} } \right)\\ = ( - 13.6 - 2.262 \times 3.007, - 13.6 + 2.262 \times 3.007)\\ = ( - 20.402, - 6.798).\\\\\end{array}\)

\(( - 20.402, - 6.798).\)