Exam 6: The Normal Probability Distribution

Suppose the amount of heating oil used annually by households in Ontario is normally distributed, with a mean of 760 litres per household per year and a standard deviation of 150 litres of heating oil per household per year. a. What is the probability that a randomly selected Ontario household uses more than 570 litres of heating oil per year? b. What is the probability that a randomly selected Ontario household uses between 680 and 1130 litres per year? c. If the members of a particular household were scared into using fuel conservation measures by newspaper accounts of the probable price of heating oil next year, and they decided they wanted to use less oil than 97.5% of all other Ontario households currently using heating oil, what is the maximum amount of oil they can use and still accomplish their conservation objective?

What is the z-score representing the third quartile of the standard normal distribution?

Loans Narrative Historical data collected at a small bank in Nova Scotia revealed that 80% of all customers applying for a loan are accepted. Suppose that 50 new loan applications are selected at random. -Refer to Loans Narrative. Find the expected value and the standard deviation of the number of loans that will be accepted by the bank.

The area between z = 0 and z = 3.50 of a standard normal curve is about 0.50.

Annual Rainfall Narrative The annual rainfall in a particular area of the country is normally distributed, with mean 100 cm and standard deviation 20 cm. -Refer to Annual Rainfall Narrative. In a given year, what is the probability that the annual rainfall will be between 105 and 125 cm?

The time it takes a student to finish a final exam is known to be normally distributed with a mean equal to 84 minutes and a standard deviation equal to 10 minutes. Given this information, the probability that it will take a randomly selected student between 75 and 90 minutes is approximately 0.0902.

Given that z is a standard normal random variable, a negative value of z indicates that the standard deviation of z is negative.

The normal random variable's density function is perfectly symmetric about a peaked central value and, thus bell shaped; and characterized by tails extending indefinitely in both directions from the centre, approaching (but never touching) the horizontal axis. All of this implies a positive probability for finding values of the random variable anywhere between minus infinity and plus infinity.

For some positive value of x, the probability that a standard normal variable is between 0 and +2x is 0.1255. Which of the following is the value of that x?

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. What is the probability that a randomly selected salmon will weigh between 3 and 5 kg?

Bike Racks Assembly Time Narrative A manufacturer of bike racks for cars claims that the assembly time x for a particular model is normally distributed, with a mean of 1 hour and a standard deviation of 0.10 hours. -Refer to Bike Racks Assembly Time Narrative. Find the probability that it takes between 0.8 hours and 1.1 hours to assemble a bike rack of this model.

Given that Z is a standard normal random variable, which of the following best describes the area to the left of a value z?

Given a normal distribution with a mean of 80 and a standard deviation of 20, which z-score would correspond to an observation of x = 50?

Tar Amounts in Cigarettes Narrative Suppose the amount of tar in cigarettes is normally distributed, with mean 3.5 mg and standard deviation 0.5 mg. -Refer to Tar Amounts in Cigarettes Narrative. "Low tar" cigarettes must have tar content below the 25th percentile of the tar content distribution. What is the value which is the 25th percentile of the tar content distribution?

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. What is the probability that a can of soup selected randomly from the entire production line will contain at most 400 mL?

Assume that x is normally distributed random variable with a mean equal to 13.4 and a standard deviation equal to 3.6. Assume also that P(x > a) = 0.05. The value of a must be 19.322.

A random variable X is normally distributed with a mean of 250 and a standard deviation of 50. Given that X = 175, its corresponding z-score is -1.50.

If a normal curve appears more peaked, what may be said of the population parameters?

Given that Z is a standard normal random variable, which of these is the value of z if the area to the right of z is 0.9066?

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 you apply your brakes, what is the probability that you will brake to a stop within 17 m or less?