Suppose x has a distribution with μ = 10 and σ = 9. (a) if a random sample of size n = 43 is drawn, find μx, σ x and p(10 ≤ x ≤ 12). (round σx to two decimal places and the probability to four decimal places.)

The probability that P(10 ≤ x ≤ 12). is 42.79%.

The z score is used to determine by how many standard deviations, the raw score is above or below the mean. The z score is given by:

From the normal distribution table, P(10 ≤ x ≤ 12) = P(0 ≤ z ≤ 1.46) = P(z < 1.46) - P(z<0) = 0.9279 - 0.5 = 42.79%.

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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.

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step-by-step explanation: