The weight of an organ in adult males has a bell-shaped distribution with a mean

Question

The weight of an organ in adult males has a bell-shaped distribution with a mean of 350 grams and a standard deviation of 45 grams determine the following (a) About 99 7% of organs will be between what weights? (b) What percentage of organs weighs between 260 grams and 440 grams? (c) What percentage of organs weighs less than 260 grams or more than 440 grams? d) What percentage of organs weighs between 260 grams and 395 grams? . Use the empirical rule to ( and grams (Use ascending order) (b) »% (Type an integer or a decimal ) (c)[ %(Type an integer or a decimal ) (d) L196 (Type an integer or decimal rounded to two decimal places as needed )

Explanation / Answer

Let X be the random variable that weight of an organ in adult males.

X ~ N(mu = 350 gm, sigma = 45 gm)

The empirical rule states that 68% will fall within the first standard deviation, 95% within the first two standard deviations, and 99.7% will fall within the first three standard deviations of the distribution's average.

a) 99.7% will fall within the first three standard deviations of the distribution's average.

So the limits are mu - 3*sigma and mu + 3*sigma.

mu - 3*sigma = 350 - 3*45 = 215

mu + 3*sigma = 350 + 3*45 = 485

99.7% will fall within 215 and 485.

b) Now we have to find percentage of organ weighs between 260 gm and 440 gm.

Lets first find P(260 < X < 440)

z-score for x = 260 and x = 440 is,

z = (260 - 350) / 45 = -2

z = (440 - 350) / 45 = 2

Now we have to find P(-2 < Z < 2)

P(-2 < Z < 2) = P(Z < 2) - P(Z < -2)

By using statistical table,

P(-2 < Z < 2) = 0.9772 - 0.0228 = 0.9545

Percentage = 0.9545*100 = 95.45%

c) Now we have to find P(X < 260 or > 440 gm)

That is we have to find P(X < 260) + P(X > 440)

z-score for x = 260 is -2

z-score for x = 440 os 2

Now we have to find P(Z < -2) + P(Z > 2)

P(Z < -2) = 0.0228

P(Z > 2) = 0.0228

P(Z < -2) + P(Z > 2) = 0.0228 + 0.0228 = 0.0455

Percentage = 0.0455*100 = 4.55%

d) To find P(260 < X < 395)

z-score for x = 395 is,

z = (395 - 350) / 45 = 1

That is we have to find P(-2 < Z < 1)

P(-2 < Z < 1) = P(Z < 1) - P(Z < -2)

= 0.8413 - 0.0228 = 0.8186

Percentage = 0.8186*100 = 81.86%

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