Respuesta :
Answer:
83.89% probability that the sample mean would be greater than 98.8 gallons
Step-by-step explanation:
To solve this question, we have to understand 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 \mu and standard deviation, which is also called standard error [tex]s = \frac{\sigma}{\sqrt{n}}[/tex]
In this problem, we have that:
The standard deviation is the square root of the variance. So
[tex]\mu = 100, \sigma = \sqrt{64} = 8, n = 44, s = \frac{8}{\sqrt{44}} = 1.21[/tex]
If 44 racing cars are randomly selected, what is the probability that the sample mean would be greater than 98.8 gallons
This probability is 1 subtracted by the pvalue of Z when X = 98.8. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{98.8 - 100}{1.21}[/tex]
[tex]Z = -0.99[/tex]
[tex]Z = -0.99[/tex] has a pvalue of 0.1611
1 - 0.1611 = 0.8389
83.89% probability that the sample mean would be greater than 98.8 gallons
