Respuesta :
Answer:
96.24% probability that a SRS of 25 students will spend an average of between 600 and 700 dollars
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 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 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.
In this problem, we have that:
[tex]\mu = 650, \sigma = 120, n = 25, s = \frac{120}{\sqrt{25}} = 24[/tex]
What is the probability that a SRS of 25 students will spend an average of between 600 and 700 dollars
This is the pvalue of Z when X = 700 subtracted by the pvalue of Z when X = 600. So
X = 700
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{700 - 650}{24}[/tex]
[tex]Z = 2.08[/tex]
[tex]Z = 2.08[/tex] has a pvalue of 0.9812
X = 600
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{600 - 650}{24}[/tex]
[tex]Z = -2.08[/tex]
[tex]Z = -2.08[/tex] has a pvalue of 0.0188
0.9812 - 0.0188 = 0.9624
96.24% probability that a SRS of 25 students will spend an average of between 600 and 700 dollars
The probability that a SRS of 25 students will spend an average of between 600 and 700 dollars is; 96.24%
We are given;
Sample size; n = 25
Population mean; μ = 650
Population standard deviation; σ = 120
We see that the sample size is less than 30, but from the Central Limit Theorem, we can say that the sampling distribution of the sample means with size n can be approximated to a normal distribution. Thus, test statistic formula is;
z = (x⁻ - μ)/(σ/√n)
Now, the probability that a SRS of 25 students will spend an average of between 600 and 700 dollars will be the difference between the p-values when; x⁻ = 600 and when x⁻ = 700
when x⁻ = 600;
z = (600 - 650)/(120/√25)
z = 2.08
when x⁻ = 700;
z = (700 - 650)/(120/√25)
z = -2.08
From z-score tables;
p-value at z = 2.08 is p = 0.9812
p-value at z = -2.08 is p = 0.0188
Thus, probability that a SRS of 25 students will spend an average of between 600 and 700 dollars is; 0.9812 - 0.0188 = 0.9624 or 96.24%
Thus, in conclusion, the probability that a SRS of 25 students will spend an average of between 600 and 700 dollars
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