The Normal Distribution and Other Continuous Distributions

These flashcards cover the key concepts and procedures related to continuous distributions, normal distribution, standardized normal distribution (Z distribution), and finding probabilities.

What is a continuous variable?

A continuous variable is one that can assume any value on a continuum, such as thickness, time, temperature, or height.

What shape does the normal distribution have?

The normal distribution has a bell-shaped curve.

What do the mean, median, and mode have in a normal distribution?

In a normal distribution, the mean, median, and mode are equal.

What determines the location and spread of a normal distribution?

The location is determined by the mean (μ) and the spread by the standard deviation (σ).

What is the standardized normal distribution?

The standardized normal distribution, known as the Z distribution, has a mean of 0 and a standard deviation of 1.

How do you convert X values to Z values?

To convert X values to Z values, subtract the mean of X and divide by its standard deviation.

What does the area under the curve of a normal distribution represent?

The area under the curve represents the probability.

How can you find the probability of an individual value in a continuous distribution?

The probability of any individual value in a continuous distribution is zero.

What does the total area under the normal curve equal?

The total area under the normal curve equals 1.0.

How can you find normal probabilities in the context of Z values?

You can find normal probabilities using the Cumulative Standardized Normal Table.

What is the procedure for finding normal probabilities?

1. Draw the normal curve for the problem in terms of X. 2. Translate X-values to Z-values. 3. Use the Cumulative Standardized Normal Table.

What does a negative Z value indicate?

A negative Z value indicates a value below the mean.

What does a Z value of 0 indicate?

A Z value of 0 indicates a value equal to the mean.

How do you find the probability of a range of values (P(a < X < b))?

Use the Z-values to find the probabilities and subtract, i.e., P(Z < b) - P(Z < a).

What is the significance of a Z value of 2.00 in the context of probability?

A Z value of 2.00 indicates that the value is two standard deviations above the mean in a standard normal distribution.

How do you determine an X value from a known probability?

1. Find the Z value for the known probability. 2. Convert to X units using the formula.

If the mean download time is 18.0 seconds and the standard deviation is 5.0 seconds, how do you find the probability of download times less than 18.6 seconds?

Convert 18.6 to a Z value, then use the Cumulative Standardized Normal Table to find the probability.

What is the Z value when calculating the probability for X > 18.6 seconds?

The Z value is calculated by standardizing 18.6 using the mean and standard deviation, and then finding P(Z > Z value).

What is the outcome of P(X > 18.6) if P(Z < 0.12) is 0.5478?

P(X > 18.6) = 1 - P(Z ≤ 0.12) = 1 - 0.5478 = 0.4522.

How can you calculate probabilities in the lower tail of the distribution?

By finding the corresponding Z-values and using the standard normal probabilities to calculate.