6. Normal Distribution and Continuous Random Variables

Explain why P(X < 30) should be reported as < 0.0001 if X is a normal random variable with mean 100 and standard deviation 15.

The arrival times of the bus Alex takes to work follow a normal distribution, with $\mu =12\min$ after the scheduled arrival rime & $\sigma =3\min$. If the bus is scheduled to arrive at Alex's work 10 min before opening, what is the probability that Alex arrives on time (i.e. the bus is less than 10 min late)?

The heights of adult women are approximately normally distributed with a mean of $160\mathrm{cm}$ and a standard deviation of $7\mathrm{cm}$. Find the height $x$ such that $5\%$ of women are shorter than $x$.

Give three interpretations for the area under a normal curve.

Suppose the scores earned on Professor McArthur’s third statistics exam are normally distributed with mean 64 and standard deviation 8. Professor McArthur wants to curve the exam scores as follows: The top 6% get an A, the next 14% get a B, the middle 60% get a C, the bottom 6% fail, and the rest earn a D. Any student who can determine these cut-offs earns five bonus points. Determine the cut-offs for Professor McArthur.

The ACT and SAT are two college entrance exams. The composite score on the ACT is approximately normally distributed with mean 21.1 and standard deviation 5.1. The composite score on the SAT is approximately normally distributed with mean 1026 and standard deviation 210. Suppose you scored 26 on the ACT and 1240 on the SAT. Which exam did you score better on? Justify your reasoning using the normal model.

In a binomial experiment with n trials and probability of success p, if __ ________, the binomial random variable X is approximately normal with μX = ________ and σX = ________.

Suppose X is a binomial random variable. To approximate P(X < 5), compute ________.

"In Problems 5–14, a discrete random variable is given. Assume the probability of the random variable will be approximated using the normal distribution. Describe the area under the normal curve that will be computed. For example, if we wish to compute the probability of finding at least five defective items in a shipment, we would approximate the probability by computing the area under the normal curve to the right of x = 4.5.

The probability that at least 40 households have a gas stove"

"In Problems 5–14, a discrete random variable is given. Assume the probability of the random variable will be approximated using the normal distribution. Describe the area under the normal curve that will be computed. For example, if we wish to compute the probability of finding at least five defective items in a shipment, we would approximate the probability by computing the area under the normal curve to the right of x = 4.5.

The probability that exactly eight defective parts are in the shipment

"

"In Problems 5–14, a discrete random variable is given. Assume the probability of the random variable will be approximated using the normal distribution. Describe the area under the normal curve that will be computed. For example, if we wish to compute the probability of finding at least five defective items in a shipment, we would approximate the probability by computing the area under the normal curve to the right of x = 4.5.

The probability that the number of people with blood type O-negative is between 18 and 24, inclusive"

"In Problems 5–14, a discrete random variable is given. Assume the probability of the random variable will be approximated using the normal distribution. Describe the area under the normal curve that will be computed. For example, if we wish to compute the probability of finding at least five defective items in a shipment, we would approximate the probability by computing the area under the normal curve to the right of x = 4.5.

The probability that more than 20 people want to see the marriage tax penalty abolished"

"Liars According to a USA Today “Snapshot,” 3% of Americans surveyed lie frequently. You conduct a survey of 500 college students and find that 20 of them lie frequently.

a. Compute the probability that, in a random sample of 500 college students, at least 20 lie frequently, assuming the true percentage is 3%."

Find the z-score such that the area to the right of the z-score is 0.483.

Find the value of z₀.₂₀.