Question

Which researcher is expected to have the narrowest histogram, most skewed distribution, a histogram with a roughly symmetric distribution

Three separate researchers in Unseen City want to estimate the mean radiation level in residents of Neighborhood 3, which was revealed by a government whistleblower to have been one of the neighborhoods exposed to radiation. Unbeknownst to these researchers the true distribution of radiation levels in the population of residents living in Neighborhood 3 is severely right skewed.

Researcher 1 takes 20 independent samples of size n = 5.

Researcher 2 takes 20 independent samples of size n = 15.

Researcher 3 takes 20 independent samples of size n = 50.

Each researcher makes a histogram consisting of the 20 sample mean values in their possession.

1. Which researcher is expected to have the narrowest histogram? How do you know?
2. Which researcher is expected to have the histogram with the most skewed distribution? How do you know?
3. Which researcher is most likely to obtain a histogram with a roughly symmetric distribution? How do you know?

Answer 1

(1)

Researcher 3 is expected to have the narrowest histogram. Reason: By Central Limit Theorem, the Standard Deviation of the Sampling Distribution of Sample mean = . So, as n increases, the histogram becomes narrow. For researcher 3, n= 50, maximum among the 3 Researchers.

(2)

Researcher 1 is expected to have the histogram with most skewed distribution. Reason: By Central Limit Theorem, the Standard Deviation of the Sampling Distribution of Sample mean = . So, if n is small, the histogram is highly skewed. For researcher 1, n= 5, minimum among the 3 Researchers.

(3)

Researcher 3 is expected to have the narrowest histogram. Reason: By Central Limit Theorem, the shape of the Sampling Distribution of Sample mean tends to Normal Distribution as n increases. So, as n increases, the histogram becomes roughly symmetric distribution. For researcher 3, n= 50, maximum among the 3 Researchers.