Understanding the Normal Probability Distribution

Question Answer
The total area under a Normal curve (or any probability distribution curve) equals what value? The total area under a Normal Curve (or any probability distribution)is 1.
For X, a normally distributed random variable, the area under the curve for a specified interval may be interpreted in what two ways? 1) The proportion of the population with the characteristic described by the interval of values, or 2) The probability that a randomly selected individual from the population will have the characteristic described by the interval of values
Normal Distribution A has mean = 5 and Standard Deviation = 2; Normal Distribution B has mean = 8 and Standard Deviation = 2. How do the shapes of these two distributions compare? The two distributions have the same shapes, but Normal Distribution B is shifted to the right 4 units.
Normal Distribution A has mean = 5 and Standard Deviation = 2; Normal Distribution B has mean = 5 and Standard Deviation = 4. How do the shapes of these two distributions compare? The two distributions have the same center(5), but Normal Distribution B is flatter than Normal Distribution A.
What is the mean of the Standard Normal Distribution? The mean of the Standard Normal Distribution is 0.
What is the standard deviation of the Standard Normal Distribution? The standard deviation of the Standard Normal Distribution is 1.
What is an alternative interpretation of the area under a Normal Curve over an interval? The area under the graph of a Normal curve over an interval represents the probability of observing a value of the random variable in that interval.
If the random variable X, is normally distributed with mean µ and standard deviation ?, what can be said about the distribution of the random variable Z =( X – µ )/?? The random variable Z is normally distributed with mean µ = 0 and standard deviation ? = 1. The random variable Z is said to have the STANDARD NORMAL DISTRIBUTION.
How does the area under the standard normal curve to right of z0 compare with the area under the standard normal curve to the left of z0. The area under the standard normal curve to the right of z0 = 1 – Area to the left of z0.
What does the notation z? represent? The notation z? (pronounced “z sub alpha”) is the z-score such that the area under the standard normal curve to the right of z? is ?.
Find the value of z0.025 (i.e., "z sub 0.025"). z0.025 = 1.96
What does P(a < Z < b) represent? P(a < Z < b) represents the probability a standard normal random variable is between a and b.
What does P(Z > a) represent? P(Z > a) represents the probability a standard normal random variable is greater than a.
What does P(Z < a)? P(Z < a) represents the probability a standard normal random variable is less than a.
For a normal distribution, does P(Z < a) have a different value from P(Z ? a)? Why? No, P(Z < a) = P(Z ? a), because a normal distribution is a type of continuous distribution. For any continuous random variable, the probability of observing a specific value of the random variable is 0.
Find each of the following probabilities: (a) P(Z < -0.23) (b) P(Z > 1.93) (c) P(0.65 < Z < 2.10) (a) P(Z < -0.23) = 0.4090 (b) P(Z > 1.93) = 0.0268 (c) P(0.65 < Z < 2.10) = 0.2399
It is known that the length of a certain steel rod is normally distributed with a mean of 100 cm and a standard deviation of 0.45 cm.* What is the probability that a randomly selected steel rod has a length less than 99.2 cm? P(X < 99.2) = P[Z <(99.2 – 100)/0.45] = P(Z < -1.78) = 0.0375
Suppose the combined (verbal + quantitative reasoning) score on the GRE is normally distributed with mean 1049 and standard deviation 189. What is the score of a student whose percentile rank is at the 85th percentile? x = µ + z? = 1049 + 1.04(189) = 1246 So the student's score at the 85th percentile is 1246.
What is the purpose of a Normal Probability Plot? A Normal Probability Plot is used to determine whether it is reasonable to assume that the sample data come from a Normal POPULATION.
How can you tell from a Normal Probability Plot whether it is reasonable to assume that the sample data come from a NORMAL POPULATION? If sample data is taken from a population that is normally distributed, a normal probability plot of the actual values versus the expected Z-scores will be approximately linear.
When using MINTAB statistical software to construct the Normal Probability Plot, how can you tell whether a normality assumption is valid for the population? In MINITAB, if the points plotted lie within the bounds provided in the graph, then we have reason to believe that the sample data comes from a population that is normally distributed.