(a) What is a standard normal variable? What are its properties?

(b) Find out the area under the standard normal curve for each of the following (use z-table). Sketch each one of them.

(i) between z = 0 and z = 0.78

(ii) between z = – 0.56 and z = 0

(iii) between z = – 0.43 and z = 0.78

(iv) between z = 0.44 and z = 1.50

(v) to the right of z = – 1.33.

A)

In probability and statistics, a standard normal variable is a variable that is distributed according to the standard normal distribution. The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1.

A standard normal variable is sometimes denoted by the letter Z. It can be used to standardize a random variable X that follows a normal distribution with a mean of μ and a standard deviation of σ by subtracting the mean and dividing by the standard deviation:

Z = (X - μ) / σ

The resulting standard normal variable Z follows the standard normal distribution. This can be useful for comparing the distribution of X to the standard normal distribution, or for using tables of the standard normal distribution to find probabilities for X.

For example, if X is a normally distributed random variable with a mean of 50 and a standard deviation of 10, then we can standardize X by calculating:

Z = (X - 50) / 10

Now, Z is a standard normal variable, and we can use tables of the standard normal distribution or a computer to find probabilities for X by finding probabilities for Z.

The standard normal distribution is a continuous probability distribution that is symmetric about the mean of 0 and has a bell-shaped curve. It is defined by the probability density function:

f(x) = 1/(√(2π)) * e^(-(x^2)/2)

where x is a real number.

Some properties of the standard normal distribution include:

• The mean, median, and mode are all equal to 0.
• The standard deviation is 1.
• The total area under the curve is 1, which means that the probability of a standard normal variable taking on any particular value is 0. However, the probability of a standard normal variable falling within a range of values can be found using integration.
• The curve is asymptotic to the x-axis, meaning that it approaches but never touches the x-axis.
• Approximately 68% of the values fall within one standard deviation of the mean, approximately 95% fall within two standard deviations, and approximately 99.7% fall within three standard deviations. This is known as the 68-95-99.7 rule or the ___ ________ _________ _______ ____ _____ ______ ____ ________ ____.
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