Exam 6: The Normal Probability Distribution

The z-score representing the 90th percentile of the standard normal curve is 1.28.

The total area under the normal probability distribution curve is equal to 1.

If x is a normal random variable with a mean of 1228 and a standard deviation of 120, how many standard deviations are there from 1228 to 1380?

What proportion of the data from a normal distribution is within two standard deviations from the mean?

The z-score representing the 10th percentile of the standard normal curve is -1.28.

Suppose you are given that x is a normally distributed random variable with a mean of and a standard deviation of 0.15. If P(x < 2.10) = 0.025, what is the value of ?

Given a normal random variable x with mean and standard deviation , the variance of the standard normal random variable z associated with x is 0.

A normal random variable x is standardized by expressing its value as the number of standard deviations it lies to the right or left of its mean .

The normal probability distribution is important because a large number of random variables observed in nature possess a frequency distribution that is approximately a normal probability distribution.

The mean of any normal distribution is always 0.

Soup Cans Narrative The liquid volumes contained in cans of soup produced by a company are normally distributed, with a mean of 425 mL and a standard deviation of 16 mL. -Refer to Soup Cans Narrative. If 28,390 of the cans of soup of the entire production contain at least 449 mL, how many cans of soup have been produced?

Pacific Salmon Narrative The owner of a fish market determined that the average weight for a Pacific salmon is 3.6 kg, with a standard deviation of 0.8 kg. Assume the weights of the salmon are normally distributed. -Refer to Pacific Salmon Narrative. A randomly selected salmon must weigh less than a certain number, say x, kg to be one of the bottom 20% in weight. What is the value of x?

The normal approximation to the binomial probability distribution is appropriate when the number of trials n is large and the probability of success p is near 0.05.

The shape of the normal probability distribution is determined by the population standard deviation $\sigma$ . Small values of $\sigma$ reduce the height of the curve and increase the spread; large values of $\sigma$ increase the height of the curve and reduce the spread.

A standard normal curve is a normal probability density function with a mean of 1 and a standard deviation of 0.

For any continuous random variable x, P(x = 5) = 1 is always true.

Braking Distance Narrative For a car travelling 60 kilometres per hour (km/h), the distance required to brake to a stop is normally distributed, with mean of 17 metres and a standard deviation of 2.4 metres. Suppose you are traveling 60 km/h in a residential area and a car moves abruptly into your path at a distance of 20 m. -Refer to Braking Distance Narrative. If the only way to avoid a collision is to brake to a stop, what is the probability that you will avoid the collision?

Textbook Narrative Twenty-five psychology instructors have formed a committee to pick next year's textbook, and they have narrowed their decision down to two equally good books, one with a better bibliography and references, and the other with a better format and illustrations. Since the books are considered to be equally good, we will assume the probability an instructor chooses either book is 0.5 and the instructors' decisions are made independently. -Refer to Textbook Narrative. Compare the results in the previous two questions.

Using the standard normal curve, the z-score representing the 75th percentile is 0.67.

In general, the binomial probability P(X = x) is approximated by the area under a normal curve between x - 0.5 and x + 0.5.