Question

10 assuming that heights of professional male tennis players follows a normal - shaped distribution, arrange in ascending order (smallest to largest).
i. a height with a z - score of 1
ii. a height with a percentile rank of 80 percent.
iii. a height at the third quartile (q3).
(a) i, ii, iii
(b) i, iii, ii
(c) ii, i, iii
(d) iii, i, ii
(e) iii, ii, i

Explanation:

Step1: Understand z - score and percentile

The z - score formula is \(z=\frac{x - \mu}{\sigma}\), where \(x\) is the data point, \(\mu\) is the mean and \(\sigma\) is the standard deviation. A z - score of \(z = 1\) means \(x=\mu+\sigma\).
For a normal distribution, the third quartile \(Q3\) corresponds to the \(75^{th}\) percentile.
The percentile rank formula: If \(P\) is the percentile rank, then the value \(x\) satisfies \(P=\frac{\text{Number of values less than }x}{n}\times100\) (for a sample of size \(n\)). A percentile rank of \(80\) means that \(80\%\) of the data is less than the value.

Step2: Compare the positions

Since \(Q3\) (third quartile) is the \(75^{th}\) percentile and a z - score of \(1\) (which is approximately the \(84.13^{th}\) percentile, from the standard normal table \(P(Z<1)=0.8413\)) and a percentile rank of \(80\).
We know that \(75^{th}\) percentile (\(Q3\)) \(<80^{th}\) percentile \(<84.13^{th}\) percentile (z - score of \(1\))

Answer:

C. II, I, III