Negative three sixth minus five sevenths.

The breaking strength of a rivet has a mean value of 10,000 psi and a standard deviation of 500 psi. (a) What is the probability

velikii [3]

Answer:

a) 89.05% probability that the sample mean breaking strength for a random sample of 40 rivets is between 9900 and 10,200

b) No, because one of the requirements of the central limit theorem is a sample size of at least 30.

Step-by-step explanation:

To solve this question, we have 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 of at least 30 can be approximated to a normal distribution with mean $\mu$ and standard deviation $s = \frac{\sigma}{\sqrt{n}}$

In this problem, we have that:

$\mu = 10000, \sigma = 500$

(a) What is the probability that the sample mean breaking strength for a random sample of 40 rivets is between 9900 and 10,200?

Here we have $n = 40, s = \frac{500}{\sqrt{40}} = 79.06$

This probability is the pvalue of Z when X = 10200 subtracted by the pvalue of Z when X = 9900. So

X = 10200

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

By the Central Limit Theorem

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

$Z = \frac{10200 - 10000}{79.06}$

$Z = 2.53$

$Z = 2.53$ has a pvalue of 0.9943.

X = 9900

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

$Z = \frac{9900 - 10000}{79.06}$

$Z = -1.26$

$Z = -1.26$ has a pvalue of 0.1038.

0.9943 - 0.1038 = 0.8905

89.05% probability that the sample mean breaking strength for a random sample of 40 rivets is between 9900 and 10,200

(b) If the sample size had been 15 rather than 40, could the probability requested in part (a) be calculated from the given information?

No, because one of the requirements of the central limit theorem is a sample size of at least 30.

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To keep cylindrical flower vases from spilling, a designer is planning to enlarge the original area of the base by 25%. The new

spin [16.1K]

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Step-by-step explanation:

Volume of cylinder formula is $V=\pi r^2 h$

Area of base is $\pi r^2$ , if this area is increased by 25%, the new area of base would be:

$\pi r^2 + \frac{25}{100}(\pi r^2)\\=\pi r^2 + (0.25)(\pi r^2)\\=1.25\pi r^2$

Since volume would be same, we can make the height as:

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Thus, we can see that the height needs to be divided by 1.25, which we can write in fractional form as:

$\frac{h}{1.25}\\=\frac{h}{\frac{5}{4}}\\=h*\frac{4}{5}\\=h*(0.8)$

Thus, height needs to be $\frac{4}{5}$ of original (or 80% of original)

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