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Volgvan
3 years ago
12

What is the length of MN¯¯¯¯¯¯¯ ? Round to the nearest tenth of a unit.

Mathematics
2 answers:
ch4aika [34]3 years ago
8 0

Answer: 8.5

Step-by-step explanation:

morpeh [17]3 years ago
3 0

Answer

Find out the length of MN .

To prove

Formula

Distance\ formula = \sqrt{{(x_{2} - x_{1})^{2} +{(y_{2} - y_{1})^{2} }

Two points be M (2, -2) and N(8,4) .

Put in the formula

Distance\ formula = \sqrt{{(8 - 2)^{2} +{(4 - (-2))^{2} }

Solving the above

Distance\ formula = \sqrt{{6^{2} +{6^{2} }

Distance\ formula = \sqrt{36+36}

Distance\ formula = \sqrt{2\times36}

Distance\ formula = \sqrt{72}

Distance formula = 8.5 unit (approx)

Therefore the length of MN is  8.5 unit (approx) .

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Rockwell hardness of pins of a certain type is known to have a mean value of 50 and a standard deviation of 1.8. (Round your ans
Alenkinab [10]

Answer:

a) 0.011 = 1.1% probability that the sample mean hardness for a random sample of 17 pins is at least 51

b) 0.0001 = 0.1% probability that the sample mean hardness for a random sample of 45 pins is at least 51

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 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.

In this problem, we have that:

\mu = 50, \sigma = 1.8

(a) If the distribution is normal, what is the probability that the sample mean hardness for a random sample of 17 pins is at least 51?

Here n = 17, s = \frac{1.8}{\sqrt{17}} = 0.4366

This probability is 1 subtracted by the pvalue of Z when X = 51. So

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

By the Central Limit Theorem

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

Z = \frac{51 - 50}{0.4366}

Z = 2.29

Z = 2.29 has a pvalue of 0.9890

1 - 0.989 = 0.011

0.011 = 1.1% probability that the sample mean hardness for a random sample of 17 pins is at least 51

(b) What is the (approximate) probability that the sample mean hardness for a random sample of 45 pins is at least 51?

Here n = 17, s = \frac{1.8}{\sqrt{45}} = 0.2683

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

Z = \frac{51 - 50}{0.0.2683}

Z = 3.73

Z = 3.73 has a pvalue of 0.9999

1 - 0.9999 = 0.0001

0.0001 = 0.1% probability that the sample mean hardness for a random sample of 45 pins is at least 51

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