Probabilities/ states question

We are told that the 'maximum recorded wind speed' is normally distributed with mean $\mu = 168$ mph and stardard deviation $\sigma = 7$ mph. If we let $X$ be a random variable with the mean and standard deviation specified above, we can write:

$$X \sim \mathrm{Normal}(168, 7)$$

Our two questions are as follows:

1. What is the probability that a future hirricane will have a max wind speed between 190 and 210 mph?

We can write this mathematically as:

$$P(190 \leq X \leq 210) = \Phi\left(\frac{210 - 168}{7}\right) - \Phi\left(\frac{190 - 168}{7}\right)$$

Where $\Phi$ represents the cumulative distribution function of the standard normal distribution. The quantities in the parentheses are our values as $z$-scores, so we get:

$$P(190 \leq X \leq 210) = \Phi\left(6\right) - \Phi\left(3.143\right)$$

We can look up these values in a $z$-score table or use a calculator to get:

$$\boxed{P(190 \leq X \leq 210) = 0.00084}$$

So there is about a $0.084\%$ chance of a given future category 5 hurricant to have max wind speeds between 190 and 210 mph.

2. What is the probability that a future catastrophic hurricane will have max wind speeds less than 160 mph?

We can write this as:

$$P(X \leq 160) = \Phi\left(\frac{160 - 168}{7}\right) = \Phi(-1.143)$$

Using a calculator we get:

$$\boxed{P(X \leq 160) = 0.127}$$

So there is a $12.7\%$ chance that a future catastrophic hurricane has wind speeds less than 160 mph.

Note: the wording for this question is a little ambiguous. In the problem set up, it states the given normal distribution is for "Category 5" hurricanes, but the questions refer to "catastrophic Hurricanes" -- I assume these are intended to be the same thing. If so, the question of whether a future storm will have wind speed less than 160 is somewhat odd, because the beginning of the prompt defines Category 5 hurricanes as having windspeeds of at least 157 mph. So looking at hurricanes with less than 160 mph winds will include hurricanes that are, by definition, not category 5. This is more of an interpretation issue than a practical one, you get similar answers regardless, I just figured it was worth pointing out.