Question

A company sells sunscreen in 400 milliliter (ml) tubes. In fact, the amount of lotion in a tube varies according to a normal distribution with mean μ=396 ml and standard deviation σ=7 ml. Suppose a store which sells this sunscreen advertises a sale for 6 tubes for the price of 5. Consider the average amount of lotion from a sample of 6 tubes of sunscreen. Find:

(a) the standard deviation of the average, ð‘¥Â¯ .
(b) the probability that the average amount of sunscreen from 6 tubes will be less than 388 ml.

Answer 1

a) The standard deviation of the average amount of sunscreen for a sample of 6 tubes is 2.86 ml.

b) The probability that the average amount from 6 tubes is less than 388 ml is approximately 0.26%.

Step-by-step explanation:

When dealing with the average (also called the mean) of a set of samples from a normal distribution, it's important to understand how the standard deviation of this average is calculated. Given a population standard deviation (σ) and a sample size (n), the standard deviation of the sample mean is equal to σ/√n. Applying this formula to the sunscreen tubes:

• Population standard deviation (σ) = 7 ml
• Sample size (n) = 6 tubes

To find the standard deviation of the average, σ we simply divide the population standard deviation by the square root of the sample size:

Sample mean = σ/√n

Sample mean = 7 ml/√6

The sample mean ≈ 2.86 ml

For part b, we need to determine the probability that the average of 6 tubes is less than 388 ml. To do this, we find the z-score by subtracting the population mean from the sample mean and dividing by the standard deviation of the average:

z = (X - μ) / Sample mean

= (388 ml - 396 ml) / 2.86 ml

≈ -2.8

a) The corresponding probability for a z-score of -2.8 is approximately 0.0026.

b) Therefore, there's about a 0.26% chance that the average amount from 6 tubes is less than 388 ml.