What is the region within three standard deviations of the mean for the standard normal distribution?

Refer to previous exercise. a. Assume that the $6$x$miles. Find out what$x$is worth. c. Assume that the$2% of tries with the shortest tread life have a tread life of no more than $x$ miles. Find out what $x$ is worth.

For the standard normal distribution, determine the area within $1.5$ standard deviations of the mean-that is, the area between $\mu-1.5 \sigma$ and $\mu+1.5 \sigma$.

In a National Highway Traffic Safety Administration (NHTSA) report, data provided to the NHTSA by Goodyear stated that the average tread life of properly inflated automobile tires is 45,000 miles (Source: http://www.nhtsa.dot.gov/cars/rules/rulings/TPMS_FMVSS_No138/part5.5.html). Assume that the current tread life distribution for automotive tires with proper inflation has a mean of 45,000 miles and a standard deviation of 2360 miles. a. Determine the likelihood that a tire from a random sample of automobiles has a tread life between 42,000 and 46,000 miles. b. What is the likelihood that a randomly chosen car tire has less than 38,000 miles left on its tread life? c. What is the likelihood that a randomly chosen car tire would last for more than 50,000 miles? d. Determine the likelihood that a vehicle tire chosen at random has a tread life between 46,500 and 47,500 miles.

Refer to earlier exercise. In that exercise, suppose the mean is set to be 8 ounces, but the standard deviation is unknown. The cups used in the machine can hold up to $8.2$ ounces, but these cups will overflow if more than $8.2$ ounces is dispensed by the machine. What is the smallest standard deviation that can be used to guarantee that overflows will happen $3% of the time?

A variation of a roulette wheel has slots that are not of equal size. Instead, the width of any slot is proportional to the probability that a standard normal random variable $z$ takes on a value between $a$ and $(a+.1)$, where $a=-3.0,-2.9,-2.8, \ldots, 2.9,3.0$. In other words, there are slots for the intervals $(-3.0,-2.9),(-2.9,-2.8),(-2.8,-2.7)$ through $(2.9,3.0)$. There is one more slot, which denotes the likelihood that $z$ will fall outside of the range $(-3.0,3.0)$. Identify the probabilities for each. a. The ball lands in the slot representing $(.3, .4)$. b. The ball lands in any of the slots representing $(-.1, .4)$. c. In at least one out of five games, the ball lands in the slot representing $(-.1,-4)$. d. In at least 100 out of 500 games, the ball lands in the slot representing $(.4, .5)$.

Suppose you are conducting a binomial experiment that has 15 trials and the probability of success of .02. You cannot approximate the binomial distribution in this case using the normal distribution due to the sample size restrictions. Explain why there is a problem in this scenario using the empirical rule, the mean, and the standard deviation of this binomial distribution. (Note: Drawing the graph and marking the values that correspond to the empirical rule is a good way to start.)

A normal distribution with a mean of 2 minutes and a standard deviation of $.5$ minute describes how long it takes a bank teller to service a randomly chosen customer. a. What is the chance that two randomly chosen customers will each be served in less than a minute? b. What is the likelihood that more than $2.25$ worth of time will be needed to serve at least one of the four randomly chosen customers?

A charter bus company is advertising a singles outing on a bus that holds 60 passengers. The company has found that, on average, $10 \%$ of ticket holders do not show up for such trips; hence, the company routinely overbooks such trips. Assume that each passenger operates on their own. a. What is the likelihood that the bus can accommodate every ticket holder who really shows up if the company sells 65 tickets? Find the likelihood that there will be 60 or fewer passengers, in other words. b. How many tickets can the company offer and still be at least $95% confident that the bus can accommodate everyone who actually shows up?

An individual gambler intends to place a series of wagers on a roulette wheel. The 38 numbers on a roulette wheel are divided into 18 red, 18 black, and 2 green. Each of the 38 numbers has an equal chance of occurring on each spin of the wheel. The gambler will select one of the two sequences shown below. Single-number bet: The gambler will bet $5$ on a particular number before each spin. He will win a net amount of $175 if that number comes up and lose$5 otherwise. Color bet: The gambler will bet $5 on the red color before each spin. He will win a net amount of$5 if a red number comes up and lose $5 otherwise. a. Which of the two betting strategies gives the gambler a better chance of coming out ahead (winning more money than losing) after placing a series of 25 bets? b. Next, determine the likelihood of winning after placing 25$5 single-number wagers and 25 $5 color wagers. Do these findings support your hunch from section a? (Before using an approximation to find either probability, be sure to check whether it is appropriate.)

Determine the region between $\mu-\sigma$ and $\mu+\sigma$ for the standard normal distribution that is within one standard deviation of the mean.

For the usual normal curve, determine the value of $z$ so that the area a. in the left tail is . 1000 b. between 0 and $z$ is $.2291$ and $z$ is positive c. in the right tail is $\mathbf{0 5 0 0}$ d. between 0 and $z$ is $.3571$ and $z$ is negative

Applying the normal probability distribution to a. a continuous random variable b. a discrete random variable c. any random variable

Find the area under the normal distribution curve. a. to the left of $z=-1.94$ b. to the left of $z=.83$ c. to the right of $z=1.45$ d. to the right of $z=-1.65$ e. between $z=.75$ and $z=1.90$ f. between $z=-1.20$ and $z=1.55$