Answer:
47.5%
Step-by-step explanation:
We are given that the length of voicemails (v) is normally distributed with a mean (μ) of 40 seconds and standard deviation (σ) of 10 seconds.
To find the probability that a given voicemail is between 20 and 40 seconds, P(20 < v < 40), we can use the empirical rule (also known as the 68-95-99.7 rule) which states:
- Approximately 68% of the data falls within one standard deviation of the mean.
- Approximately 95% falls within two standard deviations of the mean.
- Approximately 99.7% falls within three standard deviations of the mean.
Since we're interested in the probability of a voicemail being between 20 and 40 seconds, and 20 seconds is two standard deviations below the mean, we can consider this range as covering half of the 95% of data within two standard deviations, since the normal distribution curve is symmetrical about the mean.
[tex]\sf P(20 < v < 40)=\dfrac{95\%}{2} \\\\\\P(20 < v < 40)= 47.5\%[/tex]
Therefore, the probability that a given voicemail is between 20 and 40 seconds is approximately:
[tex]\LARGE\boxed{\boxed{\sf 47.5\%}}}[/tex]
