Exam 6: The Normal Probability Distribution

For a continuous random variable x and constant a, P(x a) = P(x >a) and P(x a) = P(x < a).

Let be a z-score that is unknown but identifiable by position and area. If the area to the right of is 0.8413, the value of is 1.0.

The mean, median, and standard deviation of a normally distributed random variable are all at the same position on the horizontal axis since the distribution is symmetric.

Examples of continuous probability distributions include the normal probability distribution, the Poisson probability distribution, and the binomial probability distribution.

Suppose that the probability p of success on any trial of a binomial distribution equals 0.90. For which of the following number of trials, n, would the normal distribution provide a good approximation to the binomial distribution?

Using the standard normal curve, the area between z = -1.50 and z = 1.50 is 0.4332.

The probability density function of a uniform random variable has the shape of a non-zero horizontal line over some finite interval.

If z is a standard normal random variable, the area between z = 0.0 and z = 1.20 is 0.3849, while the area between z = 0.0 and z = 1.40 is 0.4192. What is the area between z = -1.20 and z = 1.40?

National Achievement Test Scores Narrative The scores on a national achievement test were normally distributed, with a mean of 550 and a standard deviation of 112. -Refer to National Achievement Test Scores Narrative. What percentage of those who took the examination scored higher than you?

Suppose the numbers of a particular type of bacterium in samples of 1 milliliter (mL) of drinking water tend to be approximately normally distributed, with a mean of 80 and a standard deviation of 10. What is the probability that a given 1 mL sample will contain more than 100 bacteria?

Gestation Time for Human Babies Narrative The gestation time for human babies is approximately normally distributed, with an average of 280 days and a standard deviation of 12 days. -Refer to Gestation Time for Human Babies Narrative. Would it be unusual to deliver a baby after only 6 months of gestation? Explain.

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. -Given that X is a normally distributed random variable with a mean of 50 and a standard deviation of 2, find the probability that X is between 46 and 52.

Using the standard normal table, what is the total probability to the right of z = 2.0 and to the left of z = -2.0?

Gestation Time for Human Babies Narrative The gestation time for human babies is approximately normally distributed, with an average of 280 days and a standard deviation of 12 days. -Refer to Gestation Time for Human Babies Narrative. Find the upper and lower quartiles for the gestation times.

Learning Time of Computer Software Narrative The time x a student spends learning a computer software package is normally distributed with a mean of 8 hours and a standard deviation of 1.5 hours. A student is selected at random. -Refer to Learning Time of Computer Software Narrative. What is the probability that the student spends at least 8.5 hours learning the software package?

A normal random variable x has mean 54 and standard deviation 15. Would it be unusual to observe the value x = 0? Explain your answer.

For some positive value of z, the probability that a standard normal variable is between 0 and z is 0.3770. What is the value of that z?

Blood Types Narrative In a certain population, 18% of the people have Rh-negative blood. A blood bank serving this population receives 95 blood donors on a particular day. -Refer to Blood Types Narrative. What is the probability that more than 80 of the donors are Rh-positive?

In the standard normal curve, the probability or area between z = -1.28 and z = 1.28 is 0.3997.

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 longer than 1.2 hours to assemble a bike rack of this model.