Suppose that 20% of the adult women in the United States dye or highlight their hair. We would like to know the probability that

Answer:

There is a 71.08% chance that pˆ lies between 0.17 and 0.23.

Step-by-step explanation:

We apply the binomial approximation to the normal distribution for this problem.

Binomial probability distribution

This describes the likelihood of achieving exactly x successes across n trials, considering probability p.

This can be approximated using normal distribution principles with the expected value and standard deviation.

Expected value in a binomial context is:

Standard deviation in a binomial context is:

Normal probability distribution

We can solve problems with normally distributed samples via the z-score formula.

Within a data set having mean and standard deviation , the z-score for a value X is calculated as:

The Z-score indicates how far the value is from the mean in standard deviations. After calculating the Z-score, we consult the z-score table to identify the corresponding p-value. This p-value reflects the probability that the measure is less than X, signifying X's percentile. To find the likelihood that the measure exceeds X, we subtract the p-value from 1.

For approximating between a binomial and normal distribution, we have that , .

For this scenario, we find:

. Thus,

More simply put, we seek the probability that pˆ is between 0.17 and 0.23.

This probability equals the p-value of Z at X = 200*0.23 = 46 minus the p-value of Z at X = 200*0.17 = 34. Thus:

X = 46

has a p-value of 0.8554

X = 34

has a p-value of 0.1446

0.8554 - 0.1446 = 0.7108

Thus, there is a 71.08% chance that pˆ is between 0.17 and 0.23.