Using the Empirical Rule, it is found that the desired probabilities are given as follows.
a) P(x > 158) = 0.16.
b) P(149 < x < 152) = 0.135.
<h3>What does the Empirical Rule state?</h3>
It states that, for a normally distributed random variable:
- Approximately 68% of the measures are within 1 standard deviation of the mean.
- Approximately 95% of the measures are within 2 standard deviations of the mean.
- Approximately 99.7% of the measures are within 3 standard deviations of the mean.
Additionally, considering the symmetry of the normal distribution, 50% of the measures are below the mean and 50% are above.
Item a:
158 is one standard deviation above the mean, hence the probability is given by, considering that 32% of the measures are more than 1 standard deviation from the mean:
P(x > 158) = 0.5 x 0.32 = 0.16.
Item b:
Between one and two standard deviations below the mean, hence:
P(149 < x < 152) = 0.5 x (0.95 - 0.68) = 0.5 x 0.27 = 0.135.
More can be learned about the Empirical Rule at brainly.com/question/24537145
