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meriva
3 years ago
9

In Melanie's Styling Salon, the time to complete a simple haircut is normally distributed with a mean of 25 minutes and a standa

rd deviation of 4 minutes. The slowest quarter of customers will require longer than how many minutes (to the nearest tenth) for a simple haircut?
Mathematics
1 answer:
Triss [41]3 years ago
8 0

Answer:

Longer than 27.7 minutes.

Step-by-step explanation:

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.

In this question, we have that:

\mu = 25, \sigma = 4

The slowest quarter of customers will require longer than how many minutes (to the nearest tenth) for a simple haircut?

Longer than the 100 - 25 = 75th percentile of times, which is X when Z has a pvalue of 0.75. So X when Z = 0.675. So

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

0.675 = \frac{X - 25}{4}

X - 25 = 4*0.675

X = 27.7

So

Longer than 27.7 minutes.

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3 years ago
Shear strength measurements for spot welds have been found to have standard deviation 1 0 pounds per square inch (psi). If 100 t
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Answer:

P(\mu -1< \bar X

Step-by-step explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Let X the random variable that represent the Shear strength of a population, and for this case we know the distribution for X is given by:

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And let \bar X represent the sample mean, the distribution for the sample mean is given by:

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On this case  \bar X \sim N(\mu,\frac{10}{\sqrt{100}})

We are interested on this probability

P(\mu -1

And the best way to solve this problem is using the normal standard distribution and the z score given by:

z=\frac{x-\mu}{\frac{\sigma}{\sqrt{n}}}

If we apply this formula to our probability we got this:

P(\mu -1

=P(\frac{\mu -1-\mu}{\frac{10}{\sqrt{100}}}

And we can find this probability on this way:

P(-1

And in order to find these probabilities we can find tables for the normal standard distribution, excel or a calculator.  

P(-1

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