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

answer: the answer is one of

$f=\dfrac{1}{2},~\dfrac{1}{4},~\dfrac{3}{4},~\dfrac{1}{6},~\dfrac{5}{6},~\dfrac{1}{8},~\dfrac{3}{8},~\dfrac{5}{8},~\dfrac{7}{8}.$

step-by-step explanation: we are given to write a proper fraction where the whole is divided into an even number of equal parts less than ten, and an odd number of those parts exist.

an even number less than 10 may be one of 2, 4, 6 or 8. so, the whole may be one of these four evens.

now, an odd number of these parts must exist, so the numerator must be an odd number, one of 1, 3, 5 or 7. it cannot be 9 because then the fraction will become improper.

thus, our fraction may be one of the following

$f=\dfrac{1}{2},~\dfrac{1}{4},~\dfrac{3}{4},~\dfrac{1}{6},~\dfrac{5}{6},~\dfrac{1}{8},~\dfrac{3}{8},~\dfrac{5}{8},~\dfrac{7}{8}.$

Answer from: Quest

lockers 1, 4, 9, 16, 25, 40, and 49 are open

i ran through it by the other answer is wrong

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