Respuesta :
Answer:
a) Approximately normal with mean 75 and standard deviation s = 1.
b) 0.0606
c) 0.0179
d) 0.9110
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, of at least 30, can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex]
In this problem, we have that:
[tex]\mu = 75, \sigma = 8, n = 64, s = \frac{8}{\sqrt{64}} = 1[/tex]
(a) Describe the sampling distribution of x overbar.
Approximately normal with mean 75 and standard deviation s = 1.
(b) What is Upper P (x overbar greater than 76.55 )?
This is 1 subtracted by the pvalue of Z when X = 76.55.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{76.55 - 75}{1}[/tex]
[tex]Z = 1.55[/tex]
[tex]Z = 1.55[/tex] has a pvalue of 0.9394.
1 - 0.9394 = 0.0606
(c) What is Upper P (x overbar less than or equals 72.9 )?
This is the pvalue of Z when X = 72.9. So
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{72.9 - 75}{1}[/tex]
[tex]Z = -2.1[/tex]
[tex]Z = -2.1[/tex] has a pvalue of 0.0179.
(d) What is Upper P (73.5 less than x overbar less than 77.05 )?
Pvalue of Z when X = 77.05 subtracted by the pvalue of Z when X = 73.5 So
X = 77.05
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{77.05 - 75}{1}[/tex]
[tex]Z = 2.05[/tex]
[tex]Z = 2.05[/tex] has a pvalue of 0.9778
X = 73.5
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{73.5 - 75}{1}[/tex]
[tex]Z = -1.5[/tex]
[tex]Z = -1.5[/tex] has a pvalue of 0.0668
0.9778 - 0.0668 = 0.9110
