Question

1. The mean age of the viewers of a popular quiz show is 38.3 years with a standard
deviation of 12.71 years.
a) What is the Z-score of a 25-year-old viewer?
b) What is the z-score of a 70-year-old viewer?
c) What is the z-score of a 40-year-old viewer?
d) What age range is within 1 standard deviation of the mean?
e) What age range is within 2 standard deviations of the mean?
f) What age range is within 1.3 standard deviations of the mean?

Answer 1

Answer:

a) Approximately $-1.05$.

b) Approximately $2.49$.

c) Approximately $0.13$.

d) Approximately $(25.6, \, 51.0)$.

e) Approximately $(12.9,\, 63.7)$.

f) Approximately $(21.8,\, 54.8)$.

Step-by-step explanation:

The $z$-score of a measurement $x$ gives the number of standard deviations $\sigma$ between that measurement and the mean $\mu$ of the dataset.

For example, for this dataset, it is given that $\mu = 38.3$ whereas $\sigma = 12.71$. For a $25$-year-old viewer, $x = 25$.

How far away is this observation from the mean of this population? $(x - \mu) = 25 - 38.3 = -13.3$.

The $z$-score of this observation would be the approximate number of standard deviations that corresponds to this difference:

$\displaystyle z = \frac{x - \mu}{\sigma} = \frac{25 - 38.3}{12.71} \approx -1.05$.

Similarly:

For $x = 70$, $\displaystyle z = \frac{x - \mu}{\sigma} = \frac{70 - 38.3}{12.71} \approx 2.49$.

For $x = 40$, $\displaystyle z = \frac{x - \mu}{\sigma} = \frac{40 - 38.3}{12.71} \approx 0.13$.

Separately adding and subtracting $\sigma$ from $\mu$ would give the upper bound and the lower bound of the age range within $1$ standard deviation from the mean:

• Upper bound: $\mu + \sigma = 38.3 + 12.71 \approx 51.0$.
• Lower bound: $\mu - \sigma = 38.3 - 12.71 \approx 25.6$.

Similarly, for the interval within $2$ standard deviations from the mean:

• Upper bound: $\mu + 2\, \sigma = 38.3 + 2 \times 12.71 \approx 63.7$.
• Lower bound: $\mu - 2\, \sigma = 38.3 - 2\times 12.71 \approx 12.9$.

For the interval within $1.3$ standard deviations from the mean:

• Upper bound: $\mu + 1.3\, \sigma = 38.3 + 1.3 \times 12.71 \approx 54.8$.
• Lower bound: $\mu - 2\, \sigma = 38.3 - 2\times 12.71 \approx 21.8$.