The random variable x is normally distributed with mean =1,000 and standard deviation =100. Sketch and find each of the following probabilities: P(x<1,035)
Cannot sketch, but you can get probability with this equation:
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion related to the Z score.
To find the probability P(x < 1,035) for a normally distributed random variable with a mean of 1,000 and a standard deviation of 100, we can use a standard normal distribution table or a calculator with built-in functions for the normal distribution.
Here are the steps to calculate this probability:
1. Standardize the value of 1,035 to a z-score, which measures the number of standard deviations a value is from the mean. The formula to calculate the z-score is:
z = (x - μ) / σ
where x is the value (1,035), μ is the mean (1,000), and σ is the standard deviation (100).
Plugging in the values, we get:
z = (1,035 - 1,000) / 100 = 0.35
2. Using a standard normal distribution table (Z-table) or a calculator, find the area/probability to the left of the z-score calculated in step 1. This probability represents P(x < 1,035).
Consulting the Z-table or using a calculator, we find that the probability corresponding to a z-score of 0.35 is approximately 0.6368.
Therefore, the probability P(x < 1,035) is approximately 0.6368 or 63.68%.
To sketch the probability on a normal distribution curve, you can plot the mean (1,000) on the x-axis and draw a line to the value of 1,035. Shade the area under the curve to the left of this line, representing the probability P(x < 1,035).