Answer to Question #226071 in Statistics and Probability for Carrie

Question

A random sample of eight observations from three normally distributed populations produced the

following data:

Treatments

A B C

84 78 87

79 74 80

87 81 91

85 86 77

94 86 78

89 89 79

89 69 77

83 79 78

Sample mean 86:25 80:25 80:88

Sample variance 20:79 45:07 27:27

Which one of the following statements is incorrect?

1. The grand mean is 82:46.

2. The sum squares for treatments .SST / is 173:9568.

3. The sum squares of error .SSE/ is 651:91:

4. The mean squares of error .M SE/ is 3:0433:

5. The F test statistic is 2:802:

Answer 1

the grand mean

$X_{GM}=\frac{86.25+ 80.25 +80.88}{3}\\ =\frac{247.38}{3}\\ =82.46$

It is true

The sum squares

$SST= \sum n(x-\bar{x})\\ =8(82.4-82.46)^2+8(82.25-82.46)^2+8(80.88-82.46)^2\\ =8\cdot \:0.06^2+8\left(82.25-82.46\right)^2+8\left(80.88-82.46\right)^2\\ =8\cdot \:0.06^2+8\cdot \:0.21^2+8\cdot \:1.58^2\\ =20.3528$

It is incorrect

The sum squares of error

$SSE= \sum (n-1)s^2\\ =(8-1)*20.79^2+(8-1)*45.07^2+(8-1)*27.27^2\\ =7\cdot \:20.79^2+7\cdot \:45.07^2+7\cdot \:27.27^2\\ =3025.5687+14219.1343+5205.5703\\ =22450.2733$

It is incorrect

The mean squares of error

$MSE = \frac{SSE}{3}\\ = \frac{22450.2733}{3}\\ =7483.42443$

It is incorrect

The F test statistic

$F= \frac{SST/3-1}{SSE/30-3}\\ F= \frac{20.3528/3-1}{22450.2733/30-3}\\ =\frac{\frac{20.3528}{2}}{\frac{22450.2733}{30-3}}\\ =\frac{549.5256}{44900.5466}=0.01223$

It is incorrect