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Sagot :
Answer:
0.8948 = 89.48% probability that the mean of a sample of 43 cars would differ from the population mean by less than 111 miles
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal Probability Distribution
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 z-score 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 p-value, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
The Central Limit Theorem establishes 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.
The mean number of miles between services is 4959 miles, with a standard deviation of 448 miles
This means that [tex]\mu = 4959, \sigma = 448[/tex]
Sample of 43:
This means that [tex]n = 43, s = \frac{448}{\sqrt{43}}[/tex]
What is the probability that the mean of a sample of 43 cars would differ from the population mean by less than 111 miles?
p-value of Z when X = 4959 + 111 = 5070 subtracted by the p-value of Z when X = 4959 - 111 = 4848, that is, probability the sample mean is between these two values.
X = 5070
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{5070 - 4959}{\frac{448}{\sqrt{43}}}[/tex]
[tex]Z = 1.62[/tex]
[tex]Z = 1.62[/tex] has a p-value of 0.9474
X = 4848
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{4848 - 4959}{\frac{448}{\sqrt{43}}}[/tex]
[tex]Z = -1.62[/tex]
[tex]Z = -1.62[/tex] has a p-value of 0.0526
0.9474 - 0.0526 = 0.8948
0.8948 = 89.48% probability that the mean of a sample of 43 cars would differ from the population mean by less than 111 miles
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