A variable $X$ follows a normal distribution with a mean of $\mu =85$ and a standard deviation of $\sigma =10$. Interpret .

A survey counts the number of people in a group of $200$ who support a proposal. To approximate the discrete probability using a normal distribution, describe the area under the normal curve that would be computed for the event "at most $30$ supporters".

A poll claims that $4\%$ of adults prefer electric scooters. You sample $400$ commuters and find $25$ who prefer electric scooters. Compute the probability that, in a random sample of $400$ commuters, at least $25$ prefer electric scooters assuming the true proportion is $4\%$.

Let $X\sim \text{Bin}(n,p)$. Using the normal approximation with continuity correction, which expression should be used to approximate ?

A student has final exam scores in two courses. In statistics, the final scores are approximately normally distributed with mean ${\mu }_S=78$ and standard deviation ${\sigma }_S=9$, and the student scored $88$. In calculus, the final scores are approximately normally distributed with mean ${\mu }_C=82$ and standard deviation ${\sigma }_C=6$, and the student scored $89$. On which course did the student do relatively better under the normal model?

Find the value of $z_{0.07}$ where $z_{0.07}$ satisfies $P(Z\le z_{0.07})=0.07$ for a standard normal random variable $Z$.

In a binomial experiment with $n$ trials and a success probability of $p$, when can the distribution of $X$ be approximated by a normal distribution, and what are its mean and standard deviation?

Determine the $z$-score for which the area to the right is $0.025$.

In a survey of $100$ voters each independently favor a candidate with probability $0.25$. Which area under the normal curve approximates the probability that at least $30$ voters favor the candidate?

A statistics test in Professor Gomez’s class is normally distributed with a mean of $58$ and a standard deviation of $7$. Grades are assigned as follows: top $5\%$ get an A, next $15\%$ get a B, middle $60\%$ get a C, bottom $5\%$ get an F, and the rest earn a D. What are the score cut-offs for each grade?