Mathematics, 24.04.2020 20:31 misscheoneyo

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.

Answers: 3

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Answer from: Alan0201

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 $\mu$ and standard deviation $\sigma$ , the z-score of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

It measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score, which is the percentile of X.By the Central Limit Theorem, for the sampling distribution of samples of size n, the standard deviation is $s = \frac{\sigma}{\sqrt{n}}$ .

In this problem, it is stated that:

$\mu = 447, \sigma = 5$

Item a:

Samples of 4, thus n = 4 and $s = \frac{5}{\sqrt{4}} = 2.5$ .

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:

$Z = \frac{X - \mu}{\sigma}$

By the Central Limit Theorem:

$Z = \frac{X - \mu}{s}$

$Z = \frac{441 - 447}{2.5}$

Z = -2.4

Z = -2.4 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

Answer from: woodsliuheart

(a) The standard deviation of the average $\bar x$ 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,

$\mu_{\bar x}=\mu$

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

$\sigma_{\bar x}=\frac{\sigma}{\sqrt{n}}$

(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:

$\sigma_{\bar x}=\frac{\sigma}{\sqrt{n}}=\frac{5}{\sqrt{4}}=2.5$