Question

A study was undertaken to see how accurate food labeling for calories on food that is considered "reduced calorie". The group measured the amount of calories for each item of food and then found the percent difference between measured and labeled food. The group also looked at food that was nationally advertised, regionally distributed, or locally prepared. The data is in the following table ("Calories datafile," 2013).

Table: Percent Differences Between Measured and Labeled Food

National Advertised	Regionally Distributed	Locally Prepared
2	41	15
-28	46	60
-6	2	250
8	25	145
6	39	6
-1	16.5	80
10	17	95
13	28	3
15	-3
-4	14
-4	34
-18	42
10
5
3
-7
3
-0.5
-10
6

Do the data indicate that at least two of the mean percent differences between the three groups are different? Test at the 10% level.

Let x₁ = percent difference of calories between measured and labeled reduced calorie food that is nationally advertised

Let x₂ = percent difference of calories between measured and labeled reduced calorie food that is regionally distributed

Let x₃ = percent difference of calories between measured and labeled reduced calorie food that is prepared locally.

Let ?₁ = mean percent difference of calories between measured and labeled reduced calorie foods that are nationally advertised

Let ?₂ = mean percent difference of calories between measured and labeled reduced calorie foods that are regionally distributed

Let ?₃ = mean percent difference of calories between measured and labeled reduced calorie foods that are prepared locally.

(i) Which of the following statements correctly defines the null hypothesis H_O?

A. None of the three mean percentage differences are equal

B. At least two of the mean percentage differences are not equal

C. At least two of the mean percentage differences are equal

D. All three mean percentage differences are equal

Enter letter corresponding to correct answer

Answer ___________

Let ?₁ = mean percent difference of calories between measured and labeled reduced calorie foods that are nationally advertised

Let ?₂ = mean percent difference of calories between measured and labeled reduced calorie foods that are regionally distributed

Let ?₃ = mean percent difference of calories between measured and labeled reduced calorie foods that are prepared locally.

(ii) Which of the following statements correctly defines the alternate hypothesis H_A?

A. None of the three mean percentage differences are equal

B. At least two of the mean percentage differences are not equal

C. At least two of the mean percentage differences are equal

D. All three mean percentage differences are equal

Enter letter corresponding to correct answer

Answer ___________

(iii) Enter the level of significance ? used for this test:

Enter in decimal form. Examples of correctly entered answers:  0.01    0.02    0.05    0.10

Answer ___________

(iii) Calculate sample mean and sample standard deviation for sample percent differences between measured and labeled food that is advertised nationally.

Enter sample mean in decimal form to nearest thousandth, then comma, then sample standard deviation in decimal form to nearest hundredth. Examples of correctly entered answers:

13.278,2.31

0.274,0.06

-10.301,0.79

Answer ___________

(v)  Calculate sample mean and sample standard deviation for for sample percent differences between measured and labeled food that is distributed regionally.

Enter sample mean in decimal form to nearest thousandth, then comma, then sample standard deviation in decimal form to nearest hundredth. Examples of correctly entered answers:

13.278,2.31

0.274,0.06

-10.301,0.79

Answer ___________

(vi) Calculate sample mean and sample standard deviation for sample percent differences between measured and labeled food that is prepared locally.

Enter sample mean in decimal form to nearest thousandth, then comma, then sample standard deviation in decimal form to nearest hundredth. Examples of correctly entered answers:

13.278,2.31

0.274,0.06

-10.301,0.79

Answer ___________

(vii) Determine F ratio test statistic and corresponding p-value.

Use "CTRL-click" to access link. Enter test statistic to nearest hundredth, then enter comma, then enter p-value to nearest ten-thousandth. Examples of correctly entered responses:

12.33,0.0040          

7.50,0.0001

6.77,0.5049

Answer ___________

  (viii) Comparing p-value and ? value, which is the correct decision to make for this hypothesis test?

A.  Accept H_A

B. Fail to reject H_o

C. Accept H_o

D. Reject H_o

Enter letter corresponding to correct answer.

  Answer ___________

  (ix) Select the statement that most correctly interprets the result of this test:

A. The result is not statistically significant at .10 level of significance. Sufficient evidence exists to support the claim that at least two of the mean percent differences between the three groups are different.  

B. The result is statistically significant at .10 level of significance. There is not enough evidence to support the claim that at least two of the mean percent differences between the three groups are different.  

C. The result is statistically significant at .10 level of significance. Sufficient evidence exists to support the claim that at least two of the mean percent differences between the three groups are different.  

D. The result is not statistically significant at .10 level of significance. There is not enough evidence to support the claim that at least two of the mean percent differences between the three groups are different.  

Enter letter corresponding to most correct answer

Answer ___________

Answer 1

(i) The statements which correctly defines the null hypothesis HO is,

D. All three mean percentage differences are equal

(ii) The statement which correctly defines the alternate hypothesis HA is,

B. At least two of the mean percentage differences are not equal

iii) The level of significance used for this test is 10% = 0.10

(iv) The sample mean and sample standard deviation for sample percent differences between measured and labeled food that is advertised nationally is,

0.125,10.52

(v) The sample mean and sample standard deviation for sample percent differences between measured and labeled food that is Regionally Distributed is,

25.125,16.07

(vi) The sample mean and sample standard deviation for sample percent differences between measured and labeled food that is Locally Prepared is,

81.750,83.97

(vii) Determine F ratio test statistic and corresponding p-value.

Let T_i be the total percent differences for group i, n_i be number of observations of group i.

Let G be the total test scores of all observations and N be total number of observations.

$\sum$ X² for each group and the entire data set

T1 = 2.5, T2 = 301.5 , T3 = 654

G = 2.5 + 301.5 + 654 = 958

$\sum$ X² = 2103.25 + 10417.25 + 102820 = 115340.5

SST = $\sum$ X² - G²/N = 115340.5 - 958²/40 = 92396.4

SSTR = $\sum$ T²/n - G²/N = (2.5² /20 + 301.5² /12 + 654² /8 ) - 958²/40 = 38095.9

SSE = 92396.4 - 38095.9 = 54300.5

Degree of freedom of SSTR = Number of groups - 1 = 3 - 1 = 2

Degree of freedom of SSE = Number of observations - Number of groups = 40 - 3 = 37

MSTR = SSTR / Degree of freedom of SSTR = 38095.9 / 2 = 19047.95

MSE = SSE / Degree of freedom of SSE = 54300.5 / 37 = 1467.581081

F = MSTR / MSE = 19047.95 / 1467.581081 = 12.98

Numerator and Denominator degree of freedom of F statistic is Degree of freedom of SSTR, Degree of freedom of SSE = 2, 37

P-value = P(F > 12.98, DF = 2,37) = 0.0001

The answer is

12.98,0.0001

(viii) As p-value is less than the significance level (0.10), we

D. Reject Ho

(ix) The statement that most correctly interprets the result of this test is,

C. The result is statistically significant at .10 level of significance. Sufficient evidence exists to support the claim that at least two of the mean percent differences between the three groups are different.