Respuesta :
Answer:
a) 0.4772 = 47.72% probability that the average is between 1,142 and 1,150.
b) 0.0228 = 2.28% probability that the average is greater than 1,158.
c) 0 = 0% probability that the average is less than 950.
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 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.
A certain type of automobile battery is known to last an average of 1,150 days with a standard deviation of 40 days.
This means that [tex]\mu = 1150, \sigma = 40[/tex]
Sample of 100:
This means that [tex]n = 100, s = \frac{40}{\sqrt{100}} = 4[/tex]
(a) The average is between 1,142 and 1,150.
This is the pvalue of Z when X = 1150 subtracted by the pvalue of Z when X = 1142. So
X = 1150
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{1150 - 1150}{4}[/tex]
[tex]Z = 0[/tex]
[tex]Z = 0[/tex] has a pvalue of 0.5
X = 1142
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{1142 - 1150}{4}[/tex]
[tex]Z = -2[/tex]
[tex]Z = -2[/tex] has a pvalue of 0.0228
0.5 - 0.0228 = 0.4772
0.4772 = 47.72% probability that the average is between 1,142 and 1,150.
(b) The average is greater than 1,158.
This is 1 subtracted by the pvalue of Z when X = 1158. So
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{1158 - 1150}{4}[/tex]
[tex]Z = 2[/tex]
[tex]Z = 2[/tex] has a pvalue of 0.9772
1 - 0.9772 = 0.0228
0.0228 = 2.28% probability that the average is greater than 1,158.
(c) The average is less than 950.
This is the pvalue of Z when X = 950. So
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{950 - 1150}{4}[/tex]
[tex]Z = -50[/tex]
[tex]Z = -50[/tex] has a pvalue of 0
0 = 0% probability that the average is less than 950.
