according to data released by the chamber of commerce of a certain city, the weekly wages of female factory workers are normally distributed with a mean of 575 and a standard deviation of 50. Find the probability that a female factory worker selected at random from the city makes a weekly wage of 550 to 650
μ=575
σ=50
z1=(550-575)/50=-0.5
z2=(650-575)/50=+1.5
From normal distribution tables, find the probability between z=-0.5 to z=+1.5.
To find the probability that a female factory worker has a weekly wage between 550 and 650, we need to calculate the area under the normal distribution curve within this range.
Here's how you can do it:
Step 1: Standardize the values.
To work with the normal distribution, we need to convert the values of 550 and 650 into z-scores (standardized values) using the formula:
z = (x - μ) / σ
where x is the value, μ is the mean, and σ is the standard deviation.
Using this formula, the z-score for 550 is:
z1 = (550 - 575) / 50 = -0.5
The z-score for 650 is:
z2 = (650 - 575) / 50 = 1.5
Step 2: Find the probabilities associated with the z-scores.
Next, we need to find the cumulative probabilities (areas under the curve) associated with the z-scores.
Using a standard normal distribution table or a statistical calculator, find the area to the left of z1 (P(Z < -0.5)) and the area to the left of z2 (P(Z < 1.5)).
Step 3: Calculate the probability between the two z-scores.
To find the probability between the z-scores -0.5 and 1.5, we subtract the cumulative probability for -0.5 from the cumulative probability for 1.5:
P(-0.5 < Z < 1.5) = P(Z < 1.5) - P(Z < -0.5)
Step 4: Interpret the result.
The result is the probability that a female factory worker selected at random from the city makes a weekly wage of 550 to 650.
Note: If you are using a statistics software or calculator, you can directly input the values and get the probability.