Respuesta :
Answer:
0.0154 = 1.54% probability that average sleeping time for a randomly selected sample of 42 students is more than 6.5 hours per night.
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal probability distribution
When the distribution is normal, we use 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.
Central Limit Theorem
The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
In this question, we have that:
[tex]\mu = 6.3, \sigma = 0.6, n = 42, s = \frac{0.6}{\sqrt{42}} = 0.0926[/tex]
Find the probability that average sleeping time for a randomly selected sample of 42 students is more than 6.5 hours per night.
This is 1 subtracted by the pvalue of Z when X = 6.5.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{6.5 - 6.3}{0.0926}[/tex]
[tex]Z = 2.16[/tex]
[tex]Z = 2.16[/tex] has a pvalue of 0.9846
1 - 0.9846 = 0.0154
So
0.0154 = 1.54% probability that average sleeping time for a randomly selected sample of 42 students is more than 6.5 hours per night.
