A blogger observes that his blog receives an average of 3.2 visitors a day. Assuming a Poisson distribution,

1. Define the random variable and determine the mean for the given time period.

Let $X$ be the number of visitors over a two-day period.
Mean, $\lambda = 3.2 \times 2 = 6.4$
Thus, $X \sim \text{Po}(6.4)$

2. Formulate the correct probability statement and evaluate it.

The probability that the blog receives at least 5 but fewer than 10 visitors is $P(5 \le X < 10)$.
Since $X$ is a discrete random variable, this is equivalent to $P(5 \le X \le 9)$.
$P(5 \le X \le 9) = P(X \le 9) - P(X \le 4)$
Using a graphing calculator:
$P(5 \le X \le 9) \approx 0.88602 - 0.23512$
$P(5 \le X \le 9) \approx 0.65090 \approx 0.651 \text{ (to 3 s.f.)}$

1. Find the expected number of visitors in a week to set up the appropriate Poisson distribution.

Let $W$ be the number of visitors in a week (7 days).
The expected number of visitors, $\lambda = 3.2 \times 7 = 22.4$.
So, $W \sim \text{Po}(22.4)$.

2. State the suitable normal approximation and its parameters.

Since the mean $\lambda = 22.4 > 10$, $W$ can be suitably approximated by a normal distribution.
The approximate distribution is $Y \sim \text{N}(\mu, \sigma^2)$ where $\mu = 22.4$ and $\sigma^2 = 22.4$.
Thus, $Y \sim \text{N}(22.4, 22.4)$.

3. Apply continuity correction and evaluate the required probability.

We need to find the probability that the blog receives more than 30 visitors, which is $P(W > 30)$ or equivalently $P(W \ge 31)$.
Applying continuity correction for the continuous approximation:
$P(W \ge 31) \approx P(Y > 30.5)$
Using a graphing calculator with mean $22.4$ and standard deviation $\sqrt{22.4}$:
$P(Y > 30.5) \approx 0.043501$
$\approx 0.0435 \text{ (to 3 s.f.)}$

1. Identify a violated assumption of the Poisson distribution in the context of a long time period.

For a Poisson distribution to be a valid model, the events must occur at a constant average rate. Over an entire year, the average rate of 3.2 visitors per day is highly unlikely to remain constant due to real-world factors such as varying popularity, seasonal trends, holidays, or how often the blogger posts new content.