Question

# Nurses wondered if birth weights of babies are going up. They knew that the average bith weight ofa baby last year was 7.6 pounds.

Modeling data distributions
Nurses wondered if birth weights of babies are going up. They knew that the average bith weight ofa baby last year was 7.6 pounds. A random sample of 15 weights of babies atthe hospital where the nurses work gave an average birth weight of 7.9 pounds. Nurses felt thatthe birth Weights this year ware normally distributed. Which ofthe following is true about the distribution of sample means?
A. Even though the sample size is less than 30, the distribution of sample means will be normal because the population data follow a normal distribution.
B. The distribution of sample means will be normal regardless of the shape of the data in the population.
C. Since the sample size is less than 30, the population data cannot be normally distributed. Therefore, the distribution of sample means will not be normally distributed.
D. The distribution of sample means will not be normal even though the population data followed a normal distribution because the sample size is less than 30.

2021-08-11
Step 1
Given information:
Let X be the birth weight of baby.
The average birth weight of bay last year was 7.6 pounds.
The random sample of weights of babies at the hospital $$\displaystyle{\left({n}\right)}={15}$$
Average birth weight of 15 babies is $$\displaystyle{\left(\mu\right)}={7.9}$$ pounds.
The birth weight of this is year is normally distributed.
That means, $$\displaystyle{X}\sim{N}{\left({m}{e}{a}{n}={7.9},\sigma\right)}$$.
Step 2
In the given scenario, it is given that the distribution of birth weight of this year is normally distributed so it can be said that the distribution of sample means follows normal even if the sample size is less than 30.
In case, the distribution of birth weight of this year is not normal then the sample size must be 30 or more than 30 as per the central limit theorem.
Hence, the correct option is A.
Therefore, Even though the sample size is less than 30, the distribution of sample means will be normal because population data follows normal distribution.