Respuesta :
Answer:
1. c. 34%
2. b. 7.8 hours
Step-by-step explanation:
Empirical Rule:
The Empirical Rule 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.
Z-score:
Problems of normal distributions can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
In this question:
Mean of 6 hours, standard deviation of 1.5 hours.
1. Approximately what percent of people sleep between 6 and 7.5 hours per night?
6 hours = mean
7.5 hours = 6 + 1.5 = 1 standard deviation above the mean.
Approximately 68% of the measures are within 1 standard deviation of the mean. Since the normal distribution is symmetric, 34% are between one standard deviation below the mean and the mean, and 34% are between the mean(6 hours) and 1 standard deviation above the mean(7.5 hours). So the answer is 34%, given by option c.
2. If Aaron had a Z score of 1.2 how many hours did he sleep?
We have that [tex]Z = 1.2, \mu = 6, \sigma = 1.5[/tex]. We have to find X.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]1.2 = \frac{X - 6}{1.5}[/tex]
[tex]X - 6 = 1.2*1.5[/tex]
[tex]X = 7.8[/tex]
So option b.
