2. The number of violent crimes committed in a day possesses a distribution with a mean of 2.8 crimes per day and a standard deviation of 4 crimes per day. A random sample of 100 days was observed, and the sample mean number of crimes for the sample was calculated. Describe the sampling distribution of the sample means. (5 points) a.) shape unknown with mean of 2.8 and a standard deviation of 0.4​

Options :

a.) shape unknown with mean of 2.8 and a standard deviation of 0.4 b.) approximately normal with mean of 2.8 and standard deviation of 4 C.) shape unknown with mean of 2.8 and standard deviation of 4 d.) approximately normal with mean of 2.8 and standard deviation of 0.4

Answer: d.) approximately normal with mean of 2.8 and standard deviation of 0.4.

Step-by-step explanation:

Given that :

Population Mean (m) = 2.8

Population Standard deviation (s) = 4

Sample size = 100

The mean of the sampling distribution is equal to the population mean, according to the central limit theorem. As the size of the sample increases, the mean if the sample distribution converges to that if the population mean.

Hence, sample mean = 2.8

The sample standard deviation (standard error) :

Standard Error = population standard deviation / √sample size

= s/√n

= 4 / √100

= 4/10

= 0.4

Using the Central Limit Theorem, it is found that the correct description of the sampling distribution of the sample means is:

Approximately normal, with mean of 2.8 and standard deviation of 0.4.

The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].

In this problem, the population has an unknown distribution, with mean [tex]\mu = 2.8[/tex] and standard deviation [tex]\sigma = 4[/tex]

The sampling distribution of the sample means of size 100 has:

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