Question

The weekly wages of steel workers at a steel plant are normally distributed with a mean weekly wage of $940 and a standard deviation of $85. There are 1540 steel workers at this plant. a) What is the probability that a randomly selected steel worker has a weekly wage i) Of more than $850? ii) Between $910 and $960? b) What percentage of steel workers have a weekly wage of at most $820? c) Determine the total number of steel workers with a weekly wage within one standard deviation of the mean. d) Find the lowest and highest weekly wages for the middle 70% of the wage scale.

Answer 1

Given = 940 and = 85

To find the probability, we need to find the z scores.

Since n = 1,

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(a) (i) For P (X > 850) = 1 - P (X < 850), as the normal tables give us the left tailed probability only.

For P( X < 850)

Z = (850 – 940)/85 = -1.06

The probability for P(X < 850) from the normal distribution tables is = 0.1446

Therefore the required probability = 1 – 0.1446 = 0.8554

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(ii) For P (910 < X < 960) = P(X < 960) – P(X < 910)

For P( X < 960)

Z = (960 – 940)/85 = 0.24

The probability for P(X < 960) from the normal distribution tables is = 0.5948

For P( X < 910)

Z = (910 – 940)/85 = -0.35

The probability for P(X < 910) from the normal distribution tables is = 0.3632

Therefore the required probability is 0.5948 – 0.3632 = 0.2316

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(b) For P( X 820) = P(X < 820)

Z = (820 – 940)/85 = -1.41

The required probability from the normal distribution tables is = 0.0793

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(c) Within 1 standard deviation means a value below the mean 940 - 85 = 855 and a value above the mean = 940 + 85 = 1025

For P (855 < X < 1025) = P(z = 1) – P(z = -1)

The probability for z = 1 is 0.8413 and for z = -1 is 0.1587

Therefore the required probability is 0.8413 – 0.1587 = 0.6826

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(d) The middle 70%

This means that the remaining 30% will be distributed equally to the right and the left = 0.30//2 = 0.15

So the lower p value = 0.15 and the Upper p value = 1 - 0.15 = 0.85

The Z score at p = 0.15 and 0.85 are -1.0364 and +1.0364 respectively.

The Lower value: (X - 940)/85 = -1.0364. Solving for X, X = (-1.0364 * 85) + 940 = $851.9

The Upper value: (X - 940)/85 = +1.0364. Solving for X, X = (+1.0364 * 85) + 940 = $1028.1

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