The mean amount spent by a family of four on food per month is $550 with a standard deviation of $85. Assuming that the food costs are normally distributed, what is the probability that a family spends less than $410 per month?
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion that relates to your Z score.
To find the probability that a family spends less than $410 per month, we need to use the concept of Z-scores and the standard normal distribution.
Step 1: Calculate the Z-score
The Z-score is a measure of how many standard deviations an observation is from the mean. We can calculate the Z-score using the formula:
Z = (X - μ) / σ
Where:
X = the given value ($410)
μ = the mean amount spent ($550)
σ = the standard deviation ($85)
Plugging in the values, we have:
Z = (410 - 550) / 85
Z ≈ -1.647
Step 2: Find the probability associated with the Z-score
Once we have the Z-score, we can use a standard normal distribution table or a calculator to find the probability associated with it. The probability corresponds to the area under the curve to the left of the Z-score.
Using a standard normal distribution table (also called a Z-table), we can find that the cumulative probability for a Z-score of -1.647 is approximately 0.0492.
Step 3: Interpret the probability
The probability that a family spends less than $410 per month is approximately 0.0492, or 4.92%.
So, the probability that a family spends less than $410 per month is approximately 0.0492 or 4.92%.