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amm1812
3 years ago
6

A courier service company wishes to estimate the proportion of people in various states that will use its services. Suppose the

true proportion is 0.050.05. If 212212 are sampled, what is the probability that the sample proportion will differ from the population proportion by less than 0.030.03
Mathematics
1 answer:
Orlov [11]3 years ago
6 0

Answer:

95.44% probability that the sample proportion will differ from the population proportion by less than 0.03.

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.

For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean \mu = p and standard deviation s = \sqrt{\frac{p(1-p)}{n}}

In this question:

p = 0.05, n = 212, \mu = 0.05, s = \sqrt{\frac{0.05*0.95}{212}} = 0.015

What is the probability that the sample proportion will differ from the population proportion by less than 0.03?

This is the pvalue of Z when X = 0.03 + 0.05 = 0.08 subtracted by the pvalue of Z when X = 0.05 - 0.03 = 0.02. So

X = 0.08

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

By the Central Limit Theorem

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

Z = \frac{0.08 - 0.05}{0.015}

Z = 2

Z = 2 has a pvalue of 0.9772

X = 0.02

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

Z = \frac{0.02 - 0.05}{0.015}

Z = -2

Z = -2 has a pvalue of 0.0228

0.9772 - 0.0228 = 0.9544

95.44% probability that the sample proportion will differ from the population proportion by less than 0.03.

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Answer:

$37,500

Step-by-step explanation:

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We will use loss settlement formula to solve our given problem.

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Upon substituting our given values, we will get:

\text{Loss settlement}=\frac{\$50,000\times\$101,250}{\$180,000\times 75\%}

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3 years ago
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Answer:

\ln(\frac{xx + yy}{zz})^{\frac{1}{2}}

Step-by-step explanation:

Given:

1/2(ln(xx + yy) − ln(zz))

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From the properties of log function,

1)  n × ln(x) = ln(xⁿ)

and,

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applying the properties in the given equation

we get the above equation as:

\frac{1}{2}(\ln\frac{xx + yy}{zz})        

( using the property 2 we get (ln(xx + yy) − ln(zz) = \ln\frac{xx + yy}{zz}  

or

⇒ \ln(\frac{xx + yy}{zz})^{\frac{1}{2}}    ( using the property 1 i.e n × ln(x) = ln(xⁿ) )

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Step-by-step explanation:

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BlackZzzverrR [31]

Answer:

$4.49

Step-by-step explanation:

If bananas are $5.99, we need to find 75% of the price, or 25% off. There are two ways we can do this, I'll show the more detailed way. First, we multiply 5.99 by .25 to find 25% of it. We got ~1.50. This means that 25% of 5.99 is 1.50. Now, we take away 1.50 from 5.99. This gets us $4.49. That means the price of the bananas with the discount is $4.49. The other way we can do this is multiply 5.99 and .75 (because 25+75=100). This gets us 4.4925, which rounds to $4.49. I hope this helps!

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Ulleksa [173]

Answer:

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Step-by-step explanation:

To find the answer you have to add these two fractions together, and to do that you must find a common denominator.

For this problem I chose to use 20 for the denominator since it is the smallest number that both 4 and 5 have in common.

Keep in mind that, whatever you do to the denominator, you must also do to the numerator, so if you multiply 5 by 4, you must also multiply 2 by 4.

5/20 + 8/20 = 13/20

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