6. Normal Distribution and Continuous Random Variables

For the standard normal distribution, which of the following is equivalent to $p(z\ge 1.4)$?

Water Taxi Safety When a water taxi sank in Baltimore’s Inner Harbor, an investigation revealed that the safe passenger load for the water taxi was 3500 lb. It was also noted that the mean weight of a passenger was assumed to be 140 lb. Assume a “worst-case” scenario in which all of the passengers are adult men. Assume that weights of men are normally distributed with a mean of 188.6 lb and a standard deviation of 38.9 lb (based on Data Set 1 “Body Data” in Appendix B).

c. With a load limit of 3500 lb, how many male passengers are allowed if we assume the updated mean weight of 188.6 lb?

Which of the following is a characteristic of the standard normal distribution?

Small Sample Weights of M&M plain candies are normally distributed. Twelve M&M plain candies are randomly selected and weighed, and then the mean of this sample is calculated. Is it correct to conclude that the resulting sample mean cannot be considered to be a value from a normally distributed population because the sample size of 12 is too small? Explain.

Which of the following does not describe the standard normal distribution?

Given the mean of a normal distribution is $\mu$ and the standard deviation is $\sigma$, what is the mean of the corresponding standard normal distribution after standardizing the variable?

For the standard normal distribution, what is the total area under the $N(0,1)$ normal curve?

Finding Area

In Exercises 23–36, find the indicated area under the standard normal curve. If convenient, use technology to find the area.

To the left of z=1.365

For the standard normal distribution, what proportion of the distribution lies between $z=-1.75$ and $z=1.75$?

For the standard normal distribution, what is the total area under the normal curve?

A random variable Z that is normally distributed with mean μ = 0 and standard deviation σ = 1 is said to have the ____________________.

For the standard normal probability distribution, what are the mean $\mu$ and standard deviation $\sigma$?

Student's $t$ distributions are symmetric about a value of $t$. What is that value?

Find the area under the standard normal distribution to the left of a z-score of $1.21$.

Find the area under the standard normal distribution to the right of a z-score of $-0.44$.