The incomes of trainees at a local mill are normally distributed with a mean of $1100 and a standard deviation $150. What percentage of trainees earn less than $900 a month?
n P(X < 900) = P(Z < -1.33).
in the negative z score table -1.33 will be .0918, we move the decimal point to places so the answer will be 9.18%.
To find the percentage of trainees who earn less than $900 a month, we need to calculate the z-score and then use a z-table to find the corresponding percentage.
First, let's calculate the z-score using the formula:
z = (X - μ) / σ
Where:
X = Value we want to find the percentage for (in this case, $900)
μ = Mean of the distribution ($1100)
σ = Standard deviation of the distribution ($150)
z = (900 - 1100) / 150
z = -200 / 150
z = -1.33
Next, we will find the area under the curve corresponding to a z-score of -1.33. Using a standard normal distribution table (z-table), we can find the area to the left of -1.33.
Looking up -1.33 in the z-table, we find that the area to the left is 0.0918 or 9.18%.
Therefore, approximately 9.18% of trainees earn less than $900 a month.
To find the percentage of trainees who earn less than $900 a month, we can use the concept of standard deviation and the normal distribution.
First, we need to calculate the Z-score (standard score) for the value $900. The Z-score represents the number of standard deviations a specific value is from the mean.
The formula for calculating the Z-score is: Z = (X - μ) / σ
Where:
Z is the Z-score
X is the value we want to convert to a Z-score ($900 in this case)
μ is the mean ($1100 in this case)
σ is the standard deviation ($150 in this case)
Let's calculate the Z-score for $900:
Z = ($900 - $1100) / $150
Z = -200 / $150
Z = -1.3333 (rounded to four decimal places)
Now, we need to find the percentage of values that fall to the left of this Z-score using a standard normal distribution table or calculator.
Using a standard normal distribution table, we look up the Z-score of -1.3333 and find the corresponding cumulative probability (which is the percentage of values that fall below that Z-score). Let's assume it's P(Z < -1.3333) = 0.0918.
Convert the probability to a percentage:
Percentage = 0.0918 * 100
Percentage = 9.18%
Therefore, approximately 9.18% of trainees earn less than $900 a month.
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