4.E: Z-scores and the Standard Normal Distribution (Exercises)
- Page ID
- 7100
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)
- What are the two pieces of information contained in a \(z\)-score?
- Answer:
-
The location above or below the mean (from the sign of the number) and the distance in standard deviations away from the mean (from the magnitude of the number).
- A \(z\)-score takes a raw score and standardizes it into units of ________.
- Assume the following 5 scores represent a sample: 2, 3, 5, 5, 6. Transform these scores into \(z\)-scores.
- Answer:
-
\(\overline{\mathrm{X}}\)= 4.2, \(s\) = 1.64; \(z\) = -1.34, -0.73, 0.49, 0.49, 1.10
- True or false:
- All normal distributions are symmetrical
- All normal distributions have a mean of 1.0
- All normal distributions have a standard deviation of 1.0
- The total area under the curve of all normal distributions is equal to 1
- Interpret the location, direction, and distance (near or far) of the following \(z\)-scores:
- -2.00
- 1.25
- 3.50
- -0.34
- Answer:
-
- 2 standard deviations below the mean, far
- 1.25 standard deviations above the mean, near
- 3.5 standard deviations above the mean, far
- 0.34 standard deviations below the mean, near
- Transform the following \(z\)-scores into a distribution with a mean of 10 and standard deviation of 2: -1.75, 2.20, 1.65, -0.95
- Calculate \(z\)-scores for the following raw scores taken from a population with a mean of 100 and standard deviation of 16: 112, 109, 56, 88, 135, 99
- Answer:
-
\(z\) = 0.75, 0.56, -2.75, -0.75, 2.19, -0.06
- What does a \(z\)-score of 0.00 represent?
- For a distribution with a standard deviation of 20, find \(z\)-scores that correspond to:
- One-half of a standard deviation below the mean
- 5 points above the mean
- Three standard deviations above the mean
- 22 points below the mean
- Answer:
-
- -0.50
- 0.25
- 3.00
- 1.10
- Calculate the raw score for the following \(z\)-scores from a distribution with a mean of 15 and standard deviation of 3:
- 4.0
- 2.2
- -1.3
- 0.46