Use the Normal model N(1112,85) for the weight of steers a) What weight represen

Question

Use the Normal model N(1112,85) for the weight of steers

a) What weight represents the 69th percentile?

b) What weight represents the 96th percentile?

c) What's the IQR of the weights of these steers?

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Explanation / Answer

Q1)

To compute the 69th percentile, we use the formula X= + Z, and we will use the standard normal distribution table, except that we will work in the opposite direction. Previously we started with a particular "X" and used the table to find the probability. However, in this case we want to start with a 69% probability and find the value of "X" that represents it.

So we begin by going into the interior of the standard normal distribution table to find the area under the curve closest to 0.69, and from this we can determine the corresponding Z score. Once we have this we can use the equation X= + Z, because we already know that the mean and standard deviation are 1112 and 9.21, respectively.

When we go to the table, we find that the value 0.69 is not there exactly, however, the values 0.6915 there and correspond to Z values of 0.50, The exact Z value holding 69% of the values below it is 0.5 which was determined from a table of standard normal probabilities with more precision.

Using Z=0.5 the 69th percentile for men is: X = 1112 + 0.5(9.21) = 1116.605

Q2)

Again for 96th percentile>>>>>>

So we begin by going into the interior of the standard normal distribution table to find the area under the curve closest to 0.96,

When we go to the table, we find that the value 0.96 is not there exactly, however, the values0.9608 and correspond to Z values of 1.76, The exact Z value holding 96% of the values below it is 1.76 which was determined from a table of standard normal probabilities with more precision.

Using Z=1.76 the 96h percentile for men is: X = 1112 + 1.76(9.21) = 1128.21

Q3

for compute the Interr Quartile Range we havt to find the First And Third Quartile as same as above calculations>>>>

let First Quartile>>>> 25th percentile

find the area under the curve closest to 0.25, and from this we can determine the corresponding Z score. Once we have this we can use the equation X= + Z,

The values 0.2514 and correspond to Z values of -0.67, The exact Z value holding 25% of the values below it is -0.67 which was determined from a table of standard normal probabilities with more precision.

Using Z=-0.67 the 25h percentile for men is: X = 1112 -0.67(9.21) = 1105.829

Again for Third Quartile>>>>>> Q3 75th percentile

The values 0.7486 and correspond to Z values of 0.67, The exact Z value holding 75% of the values below it is 0.67 which was determined from a table of standard normal probabilities with more precision.

Using Z= 0.67 the 75h percentile for mean is: X = 1112 + 0.67(9.21) = 1118.171

IQR = Q3 - Q1

= 1118.171 - 1105.829

= 12.342

hence

69th Percentile 1116.605

96th Percentile 1128.21

Q1 1105.829

Q3 1118.171

IQR 12.342

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