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 and standard deviation , the zscore of a measure X is given by:
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 and standard deviation , a large sample size, of at least 30, can be approximated to a normal distribution with mean and standard deviation
In this problem, we have that:
(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.
By the Central Limit Theorem
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
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
has a pvalue of 0.9778
X = 73.5
has a pvalue of 0.0668
0.9778 - 0.0668 = 0.9110