China has some of the fastest trains in the world. On average, China’s trains can travel at 500 kilometers an hour, with a standard deviation of 25, and the distribution is normal.
a) On a random sample of 50 train trips in China, how many times would the speed of these trains range between 428 and 460 kilometers an hour?
b) Chinese engineers have designed a new train that can travel even faster than their current trains. The new train can travel at 550 kilometers an hour, on average, and only 1.5% of the new train’s distribution of speeds, which is also normal, would fall below the median speed of China’s current trains. What is the probability that the new train would travel faster than 620 kilometers an hour?
For a) I tried:
500-428 = 72 , 72/25 = 2.88 , p = 0.4980 (using Z=2.88)
500-450 = 40 , 20/25 = 1.60 , p = 0.4452
(0.4980-0.4452)/2 = 0.0264
But from here I don't know where to go so I'm pretty certain that the path I took is wrong. As far as b), I have no idea.
For part (a), the question is asking that if you took a random sample of the speed of $n = 50$ train trips $X_1, X_2, \ldots, X_{50}$, what is the expected number of such trips that would be between $428$ and $460$; i.e., on average, how many of the $X_i$s would satisfy $428 \le X_i \le 460$? To answer this, you have to think of each trip $X_i$ in your sample as an independent, identically distributed, normal random variable whose outcome of interest is whether it falls in the desired speed range, which occurs with some probability $p = \Pr[428 \le X_i \le 460]$. Then the number of such trips is a binomial random variable with parameters $n = 50$ and probability of "success" $p$, and its expected value is $np$.
Regarding (b), you know that the random new train speed $Y$ has a mean $\mu_Y = 550$ (whereas the old train speed has mean $\mu_X = 500$), but you are not told what the standard deviation $\sigma_Y$ of the new train speed is. Since for a normal distribution the median equals the mean, you are told that $$\Pr[Y < \mu_X] = 0.015.$$ From this relationship you should be able to determine $\sigma_Y$ by suitably standardizing $Y$.