Question

The World Health Organization has described being overweight as a global epidemic that is rapidly becoming a major public health problem. Body Mass Index (BMI) values, often used to measure obesity, show strong variability amongst countries, with New Zealand scoring at the high end. A recent study found mean BMI levels for adults aged 15 and over to be about 26 kg/m2 , with a standard deviation of about 3.5 kg/m2 , and a roughly Normal distribution. If BMI measures of a sample of 42 adults from New Zealand are collected, a) What shape should the sampling distribution of the mean have? b) What would the mean of the sampling distribution be? c) What would its standard deviation be? d) If the sample size were increased to 100, how would your answers to parts a–c change?

Answer 1

Solution :

a) If a population is normally distributed with mean μ and standard deviation σ, and a sample of size n is taken from this population, then sampling distribution of sample mean follows approximately normal distribution with mean μ and standard deviation σ/√n. It means the sampling distribution of sample mean would be bell shaped.

Let X be a random variable which represents the BMI level for adults aged 15 and over.

mean (μ) = 26 kg/m

SD (σ) = 3.5 kg/m²

And given that, X ~ N(26, 3.5²)

Since,the population is normally distributed, therefore sampling distribution of sample mean would be bell shaped.

b) The mean of the sampling distribution of mean is given by,

Hence, the mean of the sampling distribution is 26 kg/m².

c) The standard deviation of the sampling distribution would be given as follows :

Where, σ is population standard deviation and n is sample size.

We have, σ = 3.5 kg/m² and n = 42

Hence, the standard deviation of the sampling distribution is 0.540.

d) If sample size is 100, the answer in part (a) and (b) would not change, but the standard deviation of the sampling distribution would change and be given as follows :

Hence, now the standard deviation of the sampling distribution would be 0.35.