Question

News reports speak of an emerging crisis of childhood obesity in the United States. The National Health and Nutrition Examination Survey (NHANES) is a government survey run every several years recording a number of vital statistics on a random sample of Americans. A body mass index (BMI) is computed for each individual in the sample based on the individual's height and weight. Here are sample results for 8-year-old boys over the past 40 years. The table gives the sample size for 3 different surveys, the sample mean and sample standard deviation. Suppose we want to run an ANOVA to see if there is a difference in mean BMI for the three surveys. Assume each survey is a random sample and there are no extreme outliers.

Answer 1

Concepts and reason

Analysis Of Variance (ANOVA) is a statistical model used to analyse the difference between the group means. It is a way to find the significant results in an experiment. It is used when there are more than two groups.

The associated procedures developed by statistician and evolutionary biologist Ronald Fisher.

Fundamentals

The formula for the error sum of squares is as follows:

The formula of the degrees of freedom for treatment sum of squares is as follows:

The formula of the degrees of freedom for error sum of squares is as follows:

$df.=n-k $

The formula of degrees of freedom for total sum of squares is as follows:

The formula of the mean sum of squares for the treatments is as follows:

The formula for the mean sum of squares for the error is as follows:

The formula of the F statistic is as follows:

(a)

The sample size is large greater than 30. So, use ANOVA technique to see if there is any difference in mean BMI for the three surveys.

(b)

The given information is as follows:

Therefore, the sample standard deviations of each survey are:

1.The NHES II 1965 is 2.486

2.The NHANES II 1980 is 2.408

3.The NHANES II 2002 is 5.851

(c)

Null hypothesis, all group means are equal.

Alternative hypothesis, all group means are not equal.

Therefore, the mean sum of squares for groups,

The sum of squares for error is,

Degrees of freedom for error is,

The mean sum of squares for error is,

The test statistic is,

[Part c]

Ans: Part a

The sample size is large .There are no extreme outliers in the given information. Hence, use ANOVA technique to see if there is any difference in mean BMI for the three surveys.

Part b

The sample standard deviations of each survey are:

1.The NHES II 1965 is 2.486

2.The NHANES II 1980 is 2.408

3.The NHANES II 2002 is 5.851

Part c

The ANOVA table is as follows:

From Excel $=FDIST(771.1905,2,974) $ , the value of p is

Compare the p-value with the level of significance $a=0.05 $ .

Here, the p-value is lesser than the level of significance then reject the Null hypothesis.

Use p-value to conclude the decision.

Hence, conclude that all treatment means are not equal.