The mayor of the town of Quahog, Rhode Island, Adam West, conducted a study to determine the mean household income of his constituents. The study surveyed 500 households and determined that the sample mean was $30,000. The population standard deviation is $4,800. If a homeowner is randomly selected in the town of Quahog, determine the probability that the household income is between $24,336 and $31,824. Write your answer as a decimal between zero and one rounded to the ten-thousandths place.
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability between the two Z scores.
To determine the probability that the household income is between $24,336 and $31,824, we can use the standard normal distribution.
First, we need to standardize the values of $24,336 and $31,824 using the population mean and standard deviation.
To standardize a value, we subtract the population mean from the value and then divide by the population standard deviation.
Z1 = ($24,336 - $30,000) / $4,800 = -0.717
Z2 = ($31,824 - $30,000) / $4,800 = 0.381
Next, we need to look up the corresponding values of Z1 and Z2 in the standard normal distribution table.
The standard normal distribution table provides the area under the curve to the left of a given Z-score. We are interested in the area between Z1 and Z2, so we need to calculate the cumulative probability associated with Z2 and subtract the cumulative probability associated with Z1.
Using the standard normal distribution table or a calculator, we find the following probabilities:
P(Z < -0.717) = 0.2371
P(Z < 0.381) = 0.6517
To find the probability between Z1 and Z2, we subtract the cumulative probability associated with Z1 from the cumulative probability associated with Z2:
P(Z1 < Z < Z2) = P(Z < Z2) - P(Z < Z1)
= 0.6517 - 0.2371
= 0.4146
Therefore, the probability that a randomly selected household income in the town of Quahog is between $24,336 and $31,824 is approximately 0.4146.