Question

The length of country and western songs is normally distributed and has a mean of 170 seconds and a standard deviation of 40 seconds. Find the probability that a random selection of 16 songs will have mean length of 158.30 seconds or less. Assume the distribution of the lengths of the songs is normal.

Answer 1

Answer: 0.1210

Step-by-step explanation:

Given : The length of country and western songs is normally distributed with $\mu=170 \text{ seconds}$

$\sigma=40\text{ seconds}$

Sample size : $n=16$

Let x be the length of randomly selected country song.

z-score : $z=\dfrac{x-\mu}{\dfrac{\sigma}{\sqrt{n}}}$

$z=\dfrac{158.30-170}{\dfrac{40}{\sqrt{16}}}\approx-1.17$

The probability that a random selection of 16 songs will have mean length of 158.30 seconds or less by using the standard normal distribution table will be

= $P(x\leq158.30)=P(z\leq-1.17)$

$=0.1210005\approx0.1210$

Hence, the probability that a random selection of 16 songs will have mean length of 158.30 seconds or less is 0.1210