Mathematics, 11.03.2020 01:34 Ksedro1998

The fill amount of soft drink bottles is normally distributed with a mean of 2.0 litres and a standard deviation of 0.05 litres. If you select a random sample of 25 bottles, what is the probability that the sample mean will be?

a. Between 1.99 and 2.0 litres?
b. Below 1.98 litres?
c. Above 2.01 litres?
d. The probability is 99% that the sample mean will contain at least how much soft drink?

Answers: 2

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Answer from: taylorlanehart

a) 34.13% probability that the sample mean will be between 1.99 and 2 litres.

b) 2.28% probability that the sample mean will be below 1.98 litres.

c) 15.87% probability that the sample mean will be above 2.01 litres.

d) 2.02325 litres

Step-by-step explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal probability distribution:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$ , the zscore of a measure X is given by:

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

The Z-score 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. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

Central Limit Theorem:

The Central Limit Theorem estabilishes that, for a random variable X, with mean $\mu$ and standard deviation $\sigma$ , the sample means with size n can be approximated to a normal distribution with mean $\mu$ and standard deviation $s = \frac{\sigma}{\sqrt{n}}$

In this problem, we have that:

$\mu = 2, \sigma = 0.05, n = 25, s = \frac{0.05}{\sqrt{25}} = 0.01$

a. Between 1.99 and 2.0 litres?

This is the pvalue of Z when X = 2 subtracted by the pvalue of Z when X = 1.99. So

X = 2

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

By the Central Limit Theorem

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

$Z = \frac{2 - 2}{0.01}$

Z = 0

Z = 0 has a pvalue of 0.5

X = 1.99

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

$Z = \frac{1.99 - 2}{0.01}$

Z = -1

Z = -1 has a pvalue of 0.1587

0.5 - 0.1587 = 0.3413

34.13% probability that the sample mean will be between 1.99 and 2 litres.

b. Below 1.98 litres?

pvalue of Z when X = 1.98. So

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

$Z = \frac{1.98 - 2}{0.01}$

Z = -2

Z = -2 has a pvalue of 0.0228

2.28% probability that the sample mean will be below 1.98 litres.

c. Above 2.01 litres?

1 subtracted by the pvalue of Z when X = 2.01. So

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

$Z = \frac{2.01 - 2}{0.01}$

Z = 1

Z = 1 has a pvalue of 0.8413

1 - 0.8413 = 0.1587

15.87% probability that the sample mean will be above 2.01 litres.

d. The probability is 99% that the sample mean will contain at least how much soft drink?

Value of X when Z has a pvalue of 0.99. So X when Z = 2.325.

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

$2.325 = \frac{X - 2}{0.01}$

X - 2 = 0.01*2.325

X = 2.02325

Answer from: Quest

no one knows lol

step-by-step explanation:

Answer from: Quest

i dont know the answer but here's a picture of a dog

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The fill amount of soft drink bottles is normally distributed with a mean of 2.0 litres and a standa...