Mathematics, 23.02.2021 03:10, chaparro0512

Of all ladies over the age of 50, 72% report that they color their hair to hide their gray. What is the probability that under 70% color their hair in a group of 85 older women?

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answered: isaiahrodriguezsm17

0.3409 = 34.09% probability that under 70% color their hair in a group of 85 older women.

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

72% report that they color their hair to hide their gray.

This means that $\mu = p = 0.72$

What is the probability that under 70% color their hair in a group of 85 older women?

For a sample of 85, we have that $s = \sqrt{\frac{0.72*0.28}{85}} = 0.0487$

This probability is the pvalue of Z when X = 0.7. So

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

By the Central Limit Theorem

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

$Z = \frac{0.7 - 0.72}{0.0487}$

Z = -0.41

Z = -0.41 has a pvalue of 0.3409

0.3409 = 34.09% probability that under 70% color their hair in a group of 85 older women.

answered: Guest

i recommend you do what your heart desires. if you're attracted to the same gender, then you're attracted to the same gender. you can't determine what your heart wants.

answered: Guest

step-by-step explanation: