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In: Math

Heights of 10 year olds. Heights of 10 year olds, regardless of gender, closely follow a...

Heights of 10 year olds. Heights of 10 year olds, regardless of gender, closely follow a normal distribution with mean 55 inches and standard deviation 6 inches. Round all answers to two decimal places.

1. What is the probability that a randomly chosen 10 year old is shorter than 48 inches?

2. What is the probability that a randomly chosen 10 year old is between 50 and 51 inches?

3. If the shortest 10% of the class is considered very tall, what is the height cutoff for very tall? inches

4. What is the height of a 10 year old who is at the 34 th percentile? inches

Solutions

Expert Solution

Solution :

Given that ,

mean = = 55

standard deviation = = 6

1)

P(x < 48) = P((x - ) / < (48 - 55) / 6)

= P(z < -1.17)

Using standard normal table,

P(x < 48) = 0.121

Probability = 0.121

2)

P(50 < x < 51) = P((50 - 55)/ 6) < (x - ) / < (51 - 55) / 6) )

= P(-0.83 < z < -0.67)

= P(z < -0.67) - P(z < -0.83)

= 0.2514 - 0.2033 = 0.0481

Probability = 0.0481

3 )

P(Z < z) = 10% = 0.10

P(Z < -1.28) = 0.10

z = -1.28

Using z-score formula,

x = z * +

x = -1.28 * 6 + 55 = 47.32

Height cutoff = 47.32

4)

P(Z < z) = 34%

P(Z < -0.41) = 0.34

z = -0.41

Using z-score formula,

x = z * +

x = -0.41 * 6 + 55 = 52.54

34th percentile = 52.54


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