Question

in a recent year (365 days), a hospital had 5693 births. a. find the mean number of births per day. b. find the probability that in a single day, there are 18 births. (round to one decimal place as needed.) c. find the probability that in a single day, there are no births. (round to four decimal places as needed.) would 0 births in a single day be a significantly low number of births? (round to four decimal places as needed.) a. the mean number of births per day is 15.6. b. the probability that in a single day, there are 18 births is 0.0785. c. the probability that in a day, there are no births is 0. the probability is 0.05 or less. no, because the probability is greater than 0.05. yes, because the probability is 0.05 or less. yes, because the probability is greater than 0.05. no, because the probability is greater than 0.05.

Explanation:

Step1: Calculate the mean number of births per day

The total number of births in a year ($n = 5693$) and the number of days in a year ($d=365$). The mean $\mu$ is given by the formula $\mu=\frac{n}{d}$. So, $\mu=\frac{5693}{365}\approx15.6$.

Step2: Calculate the probability of a certain number of births

Assume the number of births per day follows a Poisson - distribution with parameter $\lambda=\mu = 15.6$. The probability mass function of a Poisson distribution is $P(X = k)=\frac{e^{-\lambda}\lambda^{k}}{k!}$, where $X$ is the random - variable representing the number of births per day, $\lambda$ is the mean number of births per day, $k$ is the actual number of births, and $e\approx2.71828$.
For $k = 18$:
$P(X = 18)=\frac{e^{- 15.6}(15.6)^{18}}{18!}$.
First, calculate $e^{-15.6}\approx2.71828^{-15.6}\approx0.000000377$.
$(15.6)^{18}=2.84997\times10^{20}$.
$18!=6402373705728000$.
$P(X = 18)=\frac{0.000000377\times2.84997\times10^{20}}{6402373705728000}\approx0.0785$.
For $k = 0$:
$P(X = 0)=\frac{e^{-15.6}(15.6)^{0}}{0!}=e^{-15.6}\approx0$.

Step3: Determine if 0 births is significantly low

A result is significantly low if its probability is 0.05 or less. Since $P(X = 0)\approx0\leq0.05$, 0 births in a single day is a significantly low number of births.

Answer:

a. 15.6
b. 0.0785
c. 0
No, because the probability is 0.05 or less.