Respuesta :
Answer:
The first quartile for the average percent of fat calories is 34.65.
Step-by-step explanation:
To solve this question, we use the normal probability distribution and the central limit theorem.
Normal probability distribution:
Problems of normally distributed samples are 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.
Central limit theorem:
The Central Limit Theorem estabilishes that, for a random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], a large sample size can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex]
In this problem, we have that:
[tex]\mu = 36, \sigma = 10, n = 25, s = \frac{10}{\sqrt{25}} = 2[/tex]
Find the first quartile for the average percent of fat calories.
This is the value of X when Z has a pvalue of 0.25. So X when Z = -0.675.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central limit theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]-0.675 = \frac{X - 36}{2}[/tex]
[tex]X - 36 = -0.675*2[/tex]
[tex]X = 34.65[/tex]
The first quartile for the average percent of fat calories is 34.65.
