Question

The number of violent crimes committed in a day possesses a distribution with a mean of 4.2 crimes per day and a standard deviation of 6 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 ditribution of the sample mean.

a. approximately normal with mean=4.2 and standard deviation =0.6


b. approximately normal with mean = 4.2 and standard deviation =6


c. shape unknown with mean= 4.2 and standard deviation =6


d. shape unknown with mean= 4.2 and standard deviation =0.6

Answer 1

a) is the correct answer

Explanation:

Let
`X_i` be the number of crimes on the
`i^(th)` day. each
`X_i` is a random varible with mean 4.2 and standard deviation 6, and all
`X_i` are identically distributed and independent. Since the standard deviation is 6, then the variance is 36. Summing identiaclly distributed and independent random variables will make the mean and variance add up. Lets call
`S_(100)` the sum of 100 of this random varaibles, then we have

`M(S_(100)) = 100*4.2 = 420`

`V(S_(100)) = 100*36 = 3600`

The sample mean of this 100 random variables is obtained by dividing
`S_(100)` by 100. When we multiply/divide a random variable by a number
`\lambda` , then its mean is multiplied/divided by
`\lambda` and its variance is multiplied/divided by
`{\lambda}^2` . In this case, we are dividing by 100, so the mean will be 4.2 and the variance is divided by 100² = 10000, hence, its value will be 0.36. As a result, the standard deviation of the mean sample will be the square root of 0.36, which is 0.6.

We also know, as a consecuence of the Central Limit Theorem, that the mean sample of independant, identically distributed random variables, has distribution approximately normal if the sample is big enough (100 is more than enough) . So, we conclude that

• The mean sample has approximately normal distribution
• The mean sample has mean 4.2
• The mean sample has standard deviation 0.6

The correct answer is option (a).

I hope that works for you!!