Question

Let the random variable X follow a normal distribution with muμequals=4040 and sigmaσ2equals=6464. a. Find the...

Let the random variable X follow a normal distribution with muμequals=4040 and sigmaσ2equals=6464. a. Find the probability that X is greater than 5050. b. Find the probability that X is greater than 2020 and less than 5252. c. Find the probability that X is less than 4545. d. The probability is 0.30.3 that X is greater than what​ number? e. The probability is 0.070.07 that X is in the symmetric interval about the mean between which two​ numbers?

Solution:
Given in the question
Mean = 40
Variance = 64, standard deviation = 8
Solution(a)
P(Xbar>50) = 1-P(Xbar<=50)
Z = (50-40)/8 = 1.25
From Z table we found p-value
P(Xbar>50) = 1- 0.8944 = 0.1056
Solution(b)
P(20<Xbar<52) = P(Xbar<52) - P(Xbar<20)
Z = (52-40)/8 = 1.5
Z = (20-40)/8 = -2.5
From Z table we found P-value
P(20<Xbar<52) = 0.9332 - 0.0062 = 0.927
Solution(c)
P(Xbar<45)
Z = (45-40)/8 = 0.625
From Z table we found P-value
P(Xbar<45) = 0.7357
Solution(d)
p-value = 0.7
Z-score = 0.525
0.525 = (Xbar-40)/8
4.2 = Xbar - 40
Xbar = 44.2
Solution(e)
Now given P-value = 0.3 and this is two tailed so P-value = 0.15 and 0.85
Z-Score = -1.04 and Z score = 1.04
-1.04 = (Xbar-40)/8
-8.32 = Xbar -40
Xbar = 31.68
1.04 = (Xbar-40)/8
8.32 = Xbar - 40
Xbar = 48.32

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