91影视

Denote with\({\mu _1}\)the true average of the mean body mass decrease for vegan diet and with \({\mu _2}\) for the control diet.

Given two normal distributions, the random variable (standardized)

\(T = \frac{{\bar X - \bar Y - \left( {{\mu _1} - {\mu _2}} \right)}}{{\sqrt {\frac{{S_1^2}}{m} + \frac{{S_2^2}}{n}} }}\)

has approximately students \(t\) distribution with degrees of freedom\(\nu \), where \(\nu \) is

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

has to be rounded down to the nearest integer.

The two-sample \(t\) test for testing \({H_0}:{\mu _1} - {\mu _2} = {\Delta _0}\) uses the following value of test statistic

\(t = \frac{{\bar x - \bar y - {\Delta _0}}}{{\sqrt {\frac{{s_1^2}}{m} + \frac{{s_2^2}}{n}} }}\)

For the adequate alternative hypothesis the adequate area under the \({t_\nu }\) curve is calculated is the \(P\) value.

The hypotheses of interest are \({H_0}:{\mu _1} - {\mu _2} = 1\) versus \({H_1}:{\mu _1} - {\mu _2} > 1\) (exceeds by more than 1 kg). Using the given data in the exercise, the \(t\) value is

\(t = \frac{{\bar x - \bar y - {\Delta _0}}}{{\sqrt {\frac{{s_1^2}}{m} + \frac{{s_2^2}}{n}} }} = \frac{{5.8 - 3.8 - 1}}{{\sqrt {\frac{{{{3.2}^2}}}{{32}} + \frac{{{{2.8}^2}}}{{32}}} }} = 1.33\)

and the corresponding degrees of freedom can be computed using formula

\(\nu {\rm{ }} = \frac{{{{\left( {\frac{{s_1^2}}{m} + \frac{{s_2^2}}{n}} \right)}^2}}}{{\frac{{{{\left( {s_1^2/m} \right)}^2}}}{{m - 1}} + \frac{{{{\left( {s_2^2/n} \right)}^2}}}{{n - 1}}}} = \frac{{{{\left( {\frac{{{{3.2}^2}}}{{32}} + \frac{{{{2.8}^2}}}{{32}}} \right)}^2}}}{{\frac{{{{\left( {{{3.2}^2}/32} \right)}^2}}}{{32 - 1}} + \frac{{{{\left( {{{2.8}^2}/32} \right)}^2}}}{{32 - 1}}}} = 60.93\)

The \(P\) value for one sided testing (upper) is

\(P(T > 1.33) = 0.094\)

where the value was computed using software (see the table in the appendix as well). Since

\(P(T > 1.33) = 0.094P = 0.094 > 0.05\)

do not reject null hypothesis

at \(0.05\)level. There are not enough evidence to support the claim that the true average weight loss for the vegan diet exceeds the true average weight loss for the control died more than 1 kilogram.