Answer :
The probability that the sum of the 40 values is less than 7,000 is 0.0569.
We have a random sample (40 cholesterol tests) and the sample size is relatively large (n = 40). Therefore, the sum of the 40 cholesterol values (X) can be approximated by a normal distribution with the following properties:
Mean (expected value) of the sum (μX): μX = n * μ
where n is the sample size and μ is the mean of the individual values.
μX = 40 * 180 mg/dL = 7200 mg/dL
Standard deviation of the sum (σX): σX = √n * σ
where σ is the standard deviation of the individual values.
σX = √40 * 20 mg/dL ≈ 126.49 mg/dL
We are interested in the probability that the sum of the 40 values is less than 7,000 mg/dL. We can convert this scenario into a z-score by subtracting the mean of the sum (μX) and dividing by the standard deviation of the sum (σX):
z = (X - μX) / σX = (7000 mg/dL - 7200 mg/dL) / 126.49 mg/dL ≈ -1.58
Look up the z-score (-1.58) in a standard normal distribution table. This value represents the probability that a standard normal variable will be less than -1.58. The table provides a value close to 0.0569.
Thus, the probability that the sum of the 40 values is less than 7,000 is 0.0569.
The complete question:
The distribution of results from a cholesterol test has a mean of 180 and a standard deviation of 20. A sample size of 40 is drawn randomly. Find the probability that the sum of the 40 values is less than 7,000.
