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Sagot :
Answer:
P (within $6 of 4) = 0.9164
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 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].
Mean of $139 and a standard deviation of $30.
This means that [tex]\mu = 139, \sigma = 30[/tex]
Random sample of 75 households
This means that [tex]n = 75, s = \frac{30}{\sqrt{75}} = 3.464[/tex]
75 > 30, which means that the sampling distribution is approximately normal.
Find the probability that the mean amount of electric bills for a random sample of 75 households selected from this city will be within $6 of the population mean.
This is the pvalue of Z when X = 139 + 6 = 145 subtracted by the pvalue of Z when X = 139 - 6 = 133.
X = 145
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{145 - 139}{3.464}[/tex]
[tex]Z = 1.73[/tex]
[tex]Z = 1.73[/tex] has a pvalue of 0.9582
X = 133
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{133 - 139}{3.464}[/tex]
[tex]Z = -1.73[/tex]
[tex]Z = -1.73[/tex] has a pvalue of 0.0418
0.9582 - 0.0418 = 0.9164
So
P (within $6 of 4) = 0.9164
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