Mathematics, 10.03.2021 21:50 jermarae

37% of Americans say that math is the most important subject in school. In a random sample of 400 Americans, what is the probability that between 40% and 45% will say that math is the most important subject?

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Answer from: maukjackie1974

0.1070 = 10.70% probability that between 40% and 45% will say that math is the most important subject

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 normal distributions can be 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.

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 $s = \sqrt{\frac{p(1-p)}{n}}$

37% of Americans say that math is the most important subject in school.

This means that p = 0.37

Sample of 400 Americans

This means that n = 400

Mean and standard deviation:

$\mu = p = 0.37$

$s = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.37*0.63}{400}} = 0.0241$

What is the probability that between 40% and 45% will say that math is the most important subject?

This is the pvalue of Z when X = 0.45 subtracted by the pvalue of Z when X = 0.4. So

X = 0.45

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

By the Central Limit Theorem

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

$Z = \frac{0.45 - 0.37}{0.0241}$

Z = 3.31

Z = 3.31 has a pvalue of 0.9995

X = 0.4

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

$Z = \frac{0.4 - 0.37}{0.0241}$

Z = 1.24

Z = 1.24 has a pvalue of 0.8925

0.9995 - 0.8925 = 0.1070

0.1070 = 10.70% probability that between 40% and 45% will say that math is the most important subject

Answer from: Quest

i’m here for points.

Answer from: Quest

step-by-step explanation:

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37% of Americans say that math is the most important subject in school. In a random sample of 400 Am...