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Morgarella [4.7K]
2 years ago
5

If you want to be 95​% confident of estimating the population mean to within a sampling error of plus or minus3 and the standard

deviation is assumed to be 20​, what sample size is​ required? The sample size required is
Mathematics
1 answer:
Aleksandr [31]2 years ago
5 0

Answer:

The sample size required is at least 171

Step-by-step explanation:

We have that to find our \alpha level, that is the subtraction of 1 by the confidence interval divided by 2. So:

\alpha = \frac{1-0.95}{2} = 0.025

Now, we have to find z in the Ztable as such z has a pvalue of 1-\alpha.

So it is z with a pvalue of 1-0.025 = 0.975, so z = 1.96

Now, find the margin of error M as such

M = z*\frac{\sigma}{\sqrt{n}}

In which \sigma is the standard deviation of the population and n is the size of the sample.

What sample size is​ required?

A samle size of at least n is required, in which n is found when M = 3, \sigma = 20

So

M = z*\frac{\sigma}{\sqrt{n}}

3 = 1.96*\frac{20}{\sqrt{n}}

3\sqrt{n} = 20*1.96

\sqrt{n} = \frac{20*1.96}{3}

(\sqrt{n})^{2} = (\frac{20*1.96}{3})^{2}

n = 170.7

Rouding up

The sample size required is at least 171

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A statistician calculates that 8% of Americans own a Rolls Royce. If the statistician is right, what is the probability that the
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Answer:

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Step-by-step explanation:

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Normal Probability Distribution:

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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}}

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This means that p = 0.08

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This means that n = 595

Mean and standard deviation:

\mu = p = 0.08

s = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.08*0.92}{595}} = 0.0111

What is the probability that the proportion of Rolls Royce owners in a sample of 595 Americans would differ from the population proportion by more than 3%?

Proportion above 8% + 3% = 11% or below 8% - 3% = 5%. Since the normal distribution is symmetric, these probabilities are equal, and so we find one of them and multiply by 2.

Probability the proportion is less than 5%:

P-value of Z when X = 0.05. So

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

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