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Answer:
P(10 ≤ x ≤ 12) = 0.4274
Step-by-step explanation:
Population mean = u = 10
Population Standard Deviation = = 9
Sample size = n = 43
Sample mean() is equal to the population mean. So,
Sample mean = = 10
Sample standard deviation() is equal to population standard deviation divided by square root of sample size. So,
Sample standard deviation = =
We have to find the probability that for a random sample of n = 43, the value lies between 10 and 12 i.e. P(10 ≤ x ≤ 12)
P(10 ≤ x ≤ 12) = P(x ≤ 12) - P( x ≤ 10)
We can find P(x ≤ 12 ) and P(x ≤ 10) by converting these values to z scores.
The formula for z score is:
For x =12, we get:
For x =10, we get:
So,
P(x ≤ 12) - P( x ≤ 10) = P(z ≤ 1.457) - P(z ≤ 0)
From the z table,
P(z ≤ 1.457) = 0.9274
P(z ≤ 0) = 0.5
So,
P(x ≤ 12) - P( x ≤ 10) = P(z ≤ 1.458) - P(z ≤ 0) = 0.9274 - 0.5 = 0.4274
So,
P(10 ≤ x ≤ 12) = P(x ≤ 12) - P( x ≤ 10) = 0.4274
Therefore,
The probability that for a random sample of size 43, the mean lies between 10 and 12 is 0.4274.