Question

The number of phone calls your business receives daily has mean 20.6 and standard deviation 4.4. You record the number of phone calls received over the next 100 days and compute the average. The probability that is more than 21 is:_____

a. approximately 0.9186.
b. approximately 0.1814.
c. approximately 0
d. approximately 0.464

Answer 1

The correct option is b.

Explanation:

According to the Central Limit Theorem if an unknown population is selected with mean μ and standard deviation σ and appropriately huge random samples (n > 30) are selected from this population with replacement, then the distribution of the sample means will be approximately normally.

Then, the mean of the sample means is given by,

`\mu_(\bar x)=\mu\\`

And the standard deviation of the sample means is given by,

`\sigma_(\bar x)=(\sigma)/(√(n))`

As the sample size is large, i.e. n = 100 > 30, the Central Limit Theorem can be used to approximate the sampling distribution of sample mean number of phone calls your business receives daily.

Compute the probability that the average number of phone calls your business receives daily is more than 21 as follows:

`P(\bar X>21)=P((\bar X-\mu_(\bar x))/(\sigma_(\bar x))>(21-20.6)/(4.4/√(100)))\\\\=P(Z>0.91)\\\\=1-P(Z<0.91)\\\\=1-0.81859\\\\=0.18141\\\\\approx 0.1814`

Thus, the correct option is b.