Question

The National Center for Health Statistics (NCHS) provided data on the distribution of weight (in categories) among Americans in 2002. The distribution was based on specific values of body mass index (BMI) computed as weight in kilograms over height in meters squared. Underweight was defined as BMI< 18.5, Normal weight as BMI between 18.5 and 24.9, overweight as BMI between 25 and 29.9 and obese as BMI of 30 or greater. Americans in 2002 were distributed as follows: 2% Underweight, 39% Normal Weight, 36% Overweight, and 23% Obese. Suppose we want to assess whether the distribution of BMI is different in a sample of n=3,326 participants. Using the following data, run the appropriate test at a 5% level of significance.

	BMI < 18.5	BMI 18.5-24.9	BMI 25.0-29.9	BMI ≥ 30	Total
# of participants	20	932	1374	1000	3326

Answer 1

There are four different groups based on BMI value and those four groups are Underweight, Normal, Overweight and Obese. The observed frequencies corresponding to these four groups are 20, 932, 1374 and 1000 respectively.

According to the distribution given of Americans in 2002, the estimated or the predicted frequencies are

Underweight: 3326 x 2% = 66.52 = 67

Normal: 3326 x 39% = 1297.14 = 1297

Overweight: 3326 x 36% = 1197.36 = 1197

Obese : 3326 x 23% = 764.98 = 765

The test statistic for testing the goodness of fit,

Chi-sq = Sum( (O - E)^2/E ), where O is the observed frequency and E is the expected frequency

= (20 - 67)^2 / 67 + (932 - 1297)^2 / 1297 + (1374 - 1197)^2 / 1197 + (1000 - 765)^2 / 765

= 234.05

The test statistic follows a Chi-square distribution with 3 degrees of freedom, and for Chi-square(3) distribution, the critical value for 5% level of significance 7.815 and for 1% level of significance the critical value is 11.345.

Since the observed value of the statistic is greater than the critical value, the null hypothesis is rejected and we can conclude that the given data does not support to the existing proportions of different classes.