Mathematics, 10.03.2020 19:00 dustinsampsin2486

The heights of a certain population of corn plants follow a normal distribution with mean 145 cm and standard deviation 22 cm.
1. Suppose we were to choose at random from the population samples of 16 plants each. In what percentage of the samples would the sample mean height be between 135 and 155 cm?

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Answer from: dilu76

In 93.12% of the samples the sample mean height would be between 135 and 155 cm

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 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 random variable X, with mean $\mu$ and standard deviation $\sigma$ , the sample means with size n can be approximated to a normal distribution with mean $\mu$ and standard deviation $s = \frac{\sigma}{\sqrt{n}}$

In this problem, we have that:

$\mu = 145, \sigma = 22, n = 16, s = \frac{22}{\sqrt{16}} = 5.5$

In what percentage of the samples would the sample mean height be between 135 and 155 cm?

This is the pvalue of Z when X = 155 subtracted by the pvalue of Z when X = 135. So

X = 155

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

By the Central Limit Theorem

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

$Z = \frac{155 - 145}{5.5}$

Z = 1.82

Z = 1.82 has a pvalue of 0.9656

X = 135

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

$Z = \frac{135 - 145}{5.5}$

Z = -1.82

Z = -1.82 has a pvalue of 0.0344

0.9656 - 0.0344 = 0.9312

In 93.12% of the samples the sample mean height would be between 135 and 155 cm

Answer from: Quest

Answer: the correct answer is 10step-by-step explanation: let the number be n, then the expression becomes,then“twelve more than the quotient of a number and two, decreased by four” is $(\frac{n}{2}+12)-4$ when n=4,we substitute 4 whetever we see n to get, $(\frac{4}{2}+12)-4$ we simplify to obtain, $=(2+12)-4$ we further simplify to get, $=14-4=10$ therefore the correct answer is c.

Answer from: Quest

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step-by-step explanation: