A company sells sunscreen in 450 milliliter (ml) tubes. In fact, the amount of lotion in a tube varies according to a normal distribution with mean μ=447 ml and standard deviation σ=5 ml. Suppose a store which sells this sunscreen advertises a sale for 4 tubes for the price of 3.

Consider the average amount of lotion from a SRS of 4 tubes of sunscreen and find:

(a) The standard deviation of the average, x¯ :

(b) The probability that the average amount of sunscreen from 4 tubes will be less than 441 ml.

Using the normal distribution and the central limit theorem, it is found that:

a) The standard deviation for SRS of 4 tubes is of 2.5 ml.

b) 0.0082 = 0.82% probability that the average amount of sunscreen from 4 tubes will be less than 441 ml.

In a normal distribution with mean and standard deviation , the z-score of a measure X is given by:

In this problem, it is stated that:

Item a:

Samples of 4, thus and .

The standard deviation for SRS of 4 tubes is of 2.5 ml.

Item b:

This probability is the p-value of Z when X = 441, thus:

By the Central Limit Theorem:

has a p-value of 0.0082.

0.0082 = 0.82% probability that the average amount of sunscreen from 4 tubes will be less than 441 ml.

A similar problem is given at link

(a) The standard deviation of the average is, 2.5 ml.

(b) The probability that the average amount of sunscreen from 4 tubes will be less than 441 ml is 0.0082.

Step-by-step explanation:

According to the Central Limit Theorem if we have a Normally distributed population with mean μ and standard deviation σ and random samples of any size are selected from the population with replacement, then the distribution of the sample  means will be approximately normally distributed.

Then, the mean of the distribution of sample mean is given by,

And the standard deviation of the distribution of sample mean  is given by,

(a)

The information provided is:

μ = 447 ml

σ = 5 ml

n = 4.

Since the population of amount of lotion in a tube are normally distributed, the Central limit theorem can be applied to approximate the sampling distribution of sample mean.

Then the standard deviation of the sample mean is:

Thus, the standard deviation of the average is, 2.5 ml.

(b)

Compute the value of P ( < 441) as follows:

*Use a z-table for the probability.

Thus, the probability that the average amount of sunscreen from 4 tubes will be less than 441 ml is 0.0082.

step-by-step explanation:

y = | -2(-2) |     →     y = | 4 |     →     y = 4       ⇒   (-2, 4)

y = | -2(-1) |     →     y = | 2 |     →       y =   2       ⇒   (-1, 2)    

y = | -2(0) |     →     y = | 0 |     →       y = 0         ⇒   (0, 0)

y = | -2(1) |   c →     y = | -2 |     →     y =   2         ⇒   (1, 2)    

y = | -2(2) |     →     y = | - 4 |     →     y =   4         ⇒   (2, 4)