The mean height of 2000 students at a certain college is 165 cms and SD 9 cms. What is the probability group of 5 students of that college, 3 or more students would have height more then 174cms?

Question

Answer

Given

Mean height of 2000 students at a certain college is 165 cms

Standard deviation (SD) is 9 cms

To Find

The probability that a group of 5 students of that college, 3 or more students would have a height more than 174cms

Solution

Step 1: Find the Z-score

The formula for Z-score is (X-μ)/σ, where X is the height of the student, μ is the mean height, and σ is the standard deviation

For a height of 174 cms, the Z-score would be (174-165)/9 = 1

Step 2: Find the probability of one student having a height more than 174 cms

We can use a standard normal distribution table or a calculator to find the probability

The probability of one student having a height more than 174 cms is P(Z > 1) = 0.1587

Step 3: Find the probability of three or more students having a height more than 174 cms

We can use the binomial distribution formula to find the probability

The formula is P(X ≥ r) = 1 - P(X < r),='' where='' x='' is='' the='' number='' of='' students='' with='' a='' more='' than='' 174='' cms,='' and='' r='' is='' the='' minimum='' number='' of='' students='' />

In this case, r is 3

The formula becomes P(X ≥ 3) = 1 - P(X < />

We can use a binomial distribution table or a calculator to find the probability

The probability of three or more students having a height more than 174 cms is P(X ≥ 3) = 0.0024

Step 4: Find the probability of a group of 5 students having 3 or more students with a height more than 174 cms

We can use the binomial distribution formula again to find the probability

The formula is P(X = r) = (nCr) * p^r * (1-p)^(n-r), where n is the total number of students in the group, p is the probability of one student having a height more than 174 cms, and r is the number of students with a height more than 174 cms

In this case, n is 5 and p is 0.1587

The formula becomes P(X = 3) = (5C3) * 0.1587^3 * (1-0.1587)^(5-3) = 0.0123

The probability of 4 or 5 students having a height more than 174 cms is negligible

Step 5: Final Answer

The probability that a group of 5 students of that college, 3 or more students would have a height

Answer

0.1587

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