Question

The article "'The Accuracy of Stated Energy Contents of Reduced-Energy, Commercially Prepared Foods" (J. of the Amer. Dietetic Assoc.s 2010: 116-123) presented the accompanying data on vendor-stated gross energy and measured value (both in kcal) for 10 different supermarket convenience meals):

Meal: 1 2 3 4 5 6 7 8 9 10

Stated: 180 220 190 230 200 370 250 240 80 180

Measured: 212 319 231 306 211 431 288 265 145 228

Carry out a test of hypotheses to decide whether the true average % difference from that stated differs from zero. (Note: The article stated "Although formal statistical methods do not apply to convenience samples, standard statistical tests were employed to summarize the data for exploratory purposes and to suggest directions for future studies.")

Short answer

Reject null hypothesis and conclude that true mean percentage difference between stated energy values and their measured energy value is not zero.

Step-by-step solution

To find the difference from that stated differs from zero.

Notice first that the true percentage differences needs to be tested; thus, first find all 10 percentage differences

- for every meal. The percentage can be computed using

\({x_i}{\rm{ = }}\frac{{{\rm{ Measured}}{{\rm{ }}_{\rm{i}}}{\rm{ - Stated}}{{\rm{ }}_{\rm{i}}}}}{{{\rm{ Stated}}{{\rm{ }}_{\rm{i}}}}},\quad i = 1,2, \ldots ,10.\)

\(\begin{array}{l}{x_1} = \frac{{212 - 180}}{{180}} = 0.1778 \approx 17.78\% ;\\{x_2} = \frac{{319 - 220}}{{220}} = 0.45 \approx 45\% ;\\{x_3} = \frac{{231 - 190}}{{190}} = 0.2158 \approx 21.58\% ;\\{x_4} = \frac{{306 - 230}}{{230}} = 0.3304 \approx 33.04\% ;\\{x_5} = \frac{{211 - 200}}{{200}} = 0.055 \approx 5.5\% ;\\{x_6} = \frac{{431 - 370}}{{370}} = 0.1649 \approx 16.49\% ;\\{x_7} = \frac{{288 - 250}}{{250}} = 0.152 \approx 15.2\% ;\\{x_8} = \frac{{265 - 240}}{{240}} = 0.1042 \approx 10.42\% ;\\{x_9} = \frac{{145 - 80}}{{80}} = 0.8125 \approx 81.25\% ;\\{x_{10}} = \frac{{228 - 180}}{{180}} = 0.2667 \approx 26.67\% ;\end{array}\)

To find the difference from that stated differs from zero.

The hypotheses of interest are\({H_0}:\mu = 0\)versus\({{\rm{H}}_{\rm{a}}}{\rm{:\mu }} \ne {\rm{0}}\)

where\(\mu \) is the true average of the percentage differences.

The normal probability plot suggests that the t-test can be used here (perhaps the normality is not one hundred percent sure, however use the t-test to obtain the result).

The t statistic value can be computed using formula

\(t = \frac{{\bar x - {\Delta _0}}}{{s/\sqrt n }}\)

The Sample Mean\(\bar x\)of observations\({x_1},{x_2}, \ldots ,{x_n}\)is given by

\(\begin{array}{c}\bar x = \frac{{{x_1} + {x_2} + \ldots + {x_n}}}{n}\\ = \frac{1}{n}\sum\limits_{i = 1}^n {{x_i}} \end{array}\)

The sample mean\(\bar x\)is

\(\begin{array}{c}\bar x = \frac{1}{{10}} \times (17.78 + 45 + \ldots + 26.67)\\ = 27.29\% \end{array}\)

The Sample Variance\({s^2}\)is

\({s^2} = \frac{1}{{n - 1}} \times {S_{xx}}\)

where

\(\begin{array}{c}{S_{xx}} = \sum {{{\left( {{x_i} - \bar x} \right)}^2}} \\ = \sum {x_i^2} - \frac{1}{n} \times {\left( {\sum {{x_i}} } \right)^2}\end{array}\)

The Sample Standard Deviations is

\(s = \sqrt {{s^2}} = \sqrt {\frac{1}{{n - 1}} \times {S_{xx}}} .\)

The sample standard deviation is

\(\begin{array}{l} = \sqrt {\frac{1}{{10 - 1}}\left( {{{(17.78 - 27.29)}^2} + {{(45 - 27.29)}^2} + \ldots + {{(26.67 - 27.29)}^2}} \right)} \\\\ = 22.12\% \end{array}\)

To find the test statistic

Thus, the t statistic value is

\(\begin{array}{c}t = \frac{{27.29 - 0}}{{22.12/\sqrt {10} }}\\ = 3.9\end{array}\)

To find the P-value

The degrees of freedom are \(n - 1 = 10 - 1 = 9.\) The P value for the two-sided alternative hypothesis is two times the area under the \({t_9}\) curve to the right of \(|t|\)

\(\begin{array}{c}P = 2 \times P(T > 3.9)\\ = 2 \times 0.002\\ = 0.004\end{array}\)

Where the value was computed using a software (you can use the table in the appendix of the book).

\(Since \)\(P = 0.004 < \alpha \)

Where \(\alpha \) is any reasonable significance level, reject null hypothesis

Final conclusion

Since we reject the null hypothesis, the conclusion is that the true mean percentage difference between stated energy values and their measured energy value is not zero.