Give a brief description of the variable and whether the scale of measurement is nominal, ordinal, interval, or ratio, and why.

The Measurement Principles

Nominal : People or objects with the same scale value are the same on some attribute. The values of the scale have no ‘numeric’ meaning in the way that you usually think about numbers.

Ordinal: People or objects with a higher scale value have more of some attribute. The intervals between adjacent scale values are indeterminate. Scale assignment is by the property of “greater than,” “equal to,” or “less than.

Interval: Intervals between adjacent scale values are equal with respect the attribute being measured. E.g., the difference between 8 and 9 is the same as the difference between 76 and 77.

Ratio: There is a rationale zero point for the scale. Ratios are equivalent, e.g., the ratio of 2 to 1 is the same as the ratio of 8 to 4.

Examples of the Measurement Scales

Nominal : Gender. Ethnicity. Marital Status.

Ordinal: Movie ratings (0, 1 or 2 thumbs up). U.S.D.A. quality of beef ratings (good, choice, prime). The rank order of anything.

Interval: WAIS intelligence score.

Ratio: Annual income in dollars. Length or distance in centimeters, inches, miles, etc.

Explain what it means to say that this variable is “normally distributed.” Then, define probabilityvalue and explain how a probability value of .05 in this example is related to the normal curve.

A random variableX is called “normally distributed” if X follows a normal distributionwith mean E(X)= and standard deviation . In other words, for any real number x, the probability that X is less than or equal to x is equal to

If =0 and =1, then we call this distribution the standard normal distribution, denoted by N(0,1). Its probability density functionis . The normal curve y=f(x) looks like

The probability corresponds to the area of the left red region R.

The probability corresponds to the area of the right blue region R’.

The solution rewrites equations and looks at the density functions of a normal curve.