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malfutka [58]
2 years ago
8

The cost to produce a batch of granola bars is approximately Normally distributed with a mean of $7.19 and a standard deviation

of $0.86. If a random sample of 12 batches of granola bars is selected, what is the probability that the mean cost will be more than $7.00?
0.2220

0.2339

0.7653

0.7780
Mathematics
1 answer:
Bumek [7]2 years ago
6 0

Answer:

0.7780

Step-by-step explanation:

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

Normal probability distribution

When the distribution is normal, we use 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 normally distributed random variable X, with mean \mu and standard deviation \sigma, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean \mu and standard deviation s = \frac{\sigma}{\sqrt{n}}.

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

Population:

We have that \mu = 7.19, \sigma = 0.86

Sample of 12:

n = 12, s = \frac{0.86}{\sqrt{12}} = 0.2483

What is the probability that the mean cost will be more than $7.00?

This is 1 subtracted by the pvalue of Z when X = 7. So

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

By the Central Limit Theorem

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

Z = \frac{7 - 7.19}{0.2483}

Z = -0.765

Z = -0.765 has a pvalue of 0.222

1 - 0.222 = 0.778

So the answer is 0.7780

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