Question

The number of violent crimes committed in a day possesses a distribution with a mean of 2.3 crimes per day and a standard deviation of 2 crimes per day. A random sample of 100 days was observed, and the mean. number of crimes for the sample was calculated. Describe the sampling distribution of the sample mean.

A) shape unknown with mean-2.3 and standard deviation 2
B) approximately normal with mean 2.3 and standard deviation 2
C) shape unknown with mean 2.3 and standard deviation 0.2
D) approximately normal with mean 2.3 and standard deviation 0.2

Answer 1

B) approximately normal with mean 2.3 and standard deviation 2

Explanation:

Given the following :

Mean crimes per day = 2.3

Standard deviation = 2

Number of samples = 100

Sampling distribution of the mean:

According to the central limit theorem:

Sample mean = population mean

2.3 = 2.3

The standard deviation of the sample (s) : ratio of the population standard deviation and square root of the sample size.

s = population standard deviation / √sample size

s = 2 / √100

s = 2 / 10

s = 0.2

The central limit theorem also posits that once ths sample size is large enough, the sampling of the sample mean will be approximately normal.

Hence, the distribution is approximately normal, with mean of 2.3 and standard deviation of 0.2