To determine the probability that a given voicemail length [tex]\( v \)[/tex] is between 10 and 40 seconds, we first need to understand the normal distribution. In this case, the voicemail lengths are normally distributed with a mean ([tex]\( \mu \)[/tex]) of 40 seconds and a standard deviation ([tex]\( \sigma \)[/tex]) of 10 seconds.
1. Calculate the Z-scores for the given bounds:
The Z-score is calculated using the formula:
[tex]\[
Z = \frac{(X - \mu)}{\sigma}
\][/tex]
- For the lower bound (10 seconds):
[tex]\[
Z_{\text{lower}} = \frac{(10 - 40)}{10} = \frac{-30}{10} = -3.0
\][/tex]
- For the upper bound (40 seconds):
[tex]\[
Z_{\text{upper}} = \frac{(40 - 40)}{10} = \frac{0}{10} = 0.0
\][/tex]
2. Interpret the Z-scores using the 68%-95%-99.7% rule:
We use the 68%-95%-99.7% rule to find the probability within certain standard deviations for a normally distributed variable:
- About 68% of the data lies within 1 standard deviation ([tex]\( \sigma \)[/tex]) from the mean.
- About 95% of the data lies within 2 standard deviations ([tex]\( 2\sigma \)[/tex]) from the mean.
- About 99.7% of the data lies within 3 standard deviations ([tex]\( 3\sigma \)[/tex]) from the mean.
Using these rules:
- A Z-score of -3.0 means the value is 3 standard deviations below the mean.
- A Z-score of 0.0 means the value is exactly at the mean.
3. Determine the probability based on the Z-scores:
Given that [tex]\( Z_{\text{lower}} = -3.0 \)[/tex] and [tex]\( Z_{\text{upper}} = 0.0 \)[/tex]:
- [tex]\( -3.0 \)[/tex] standard deviations is more than 2 standard deviations away from the mean.
- [tex]\( 0.0 \)[/tex] standard deviations is within 1 standard deviation from the mean.
Therefore, the range from [tex]\( Z_{\text{lower}} = -3.0 \)[/tex] to [tex]\( Z_{\text{upper}} = 0.0 \)[/tex] does not fit within the first or second standard deviations in the context of the 68%-95%-99.7% rule.
Hence, the probability that a given voicemail is between 10 and 40 seconds does not fall within the usual standard deviation ranges of the 68%-95%-99.7% rule.
So the probablity that a voicemail length is between 10 seconds and 40 seconds is "Not within the given standard deviations range in the 68%-95%-99.7% rule."
