Mathematics: 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?

1. The mean age of the viewers of a popular quiz

Ответ:

peachijmin

09.02.2022

Mathematics

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 -score of a measurement 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 -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 -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 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 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: $\mu - 2\, \sigma = 38.3 - 2\times 12.71 \approx 21.8$ .

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