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?

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 and standard deviation , the zscore of a measure X is given by:

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 and standard deviation , the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean and standard deviation .

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 and standard deviation .

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

This means that

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

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

By the Central Limit Theorem

has a pvalue of 0.3409

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

answer: 1 and 3

step-by-step explanation:

is means division by fractions

step-by-step explanation: