Understanding the Normal Probability Distribution
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| 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.
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| 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
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| 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.
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| 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.
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| What is the mean of the Standard Normal Distribution? | The mean of the Standard Normal Distribution is 0.
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| What is the standard deviation of the Standard Normal Distribution? | The standard deviation of the Standard Normal Distribution is 1.
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| 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.
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| If the random variable X, is normally distributed with mean µ and standard deviation σ, what can be said about the distribution of the random variable Z =( X – µ )/σ? | The random variable Z is normally distributed with mean µ = 0 and standard deviation σ = 1. The random variable Z is said to have the STANDARD NORMAL DISTRIBUTION.
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| 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.
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| What does the notation zα represent? | The notation zα (pronounced “z sub alpha”) is the z-score such that the area under the standard normal curve to the right of zα is α.
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| Find the value of z0.025 (i.e., "z sub 0.025"). | z0.025 = 1.96
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| What does P(a < Z < b) represent? | P(a < Z < b) represents the probability a standard normal random variable is between a and b.
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| What does P(Z > a) represent? | P(Z > a) represents the probability a standard normal random variable is greater than a.
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| What does P(Z < a)? | P(Z < a) represents the probability a standard normal random variable is less than a.
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| 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.
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| 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
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| 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
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| 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 + zσ = 1049 + 1.04(189) = 1246 So the student's score at the 85th percentile is 1246.
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| 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.
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| 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.
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| 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.
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