Using the normal distribution, it is found that there is a 0.383 = 38.3% probability that the sample proportion is between 18% and 22%.
Normal Probability Distribution
The z-score of a measure X of a normally distributed variable with mean and standard deviation is given by:
- The z-score measures how many standard deviations the measure is above or below the mean.
- Looking at the z-score table, the p-value associated with this z-score is found, which is the percentile of X.
- By the Central Limit Theorem, for a proportion p in a sample of size n, the sampling distribution of sample proportion is approximately normal with mean and standard deviation , as long as and .
In this problem, the proportion and the sample size are of p = 0.2 and n = 100, respectively, hence:
The probability that in a random sample of 100 M&M candies, there are between 18% and 22% orange candies is the p-value of Z when X = 0.22 subtracted by the p-value of Z when X = 0.18, hence:
X = 0.22:
By the Central Limit Theorem:
Z = 0.5
Z = 0.5 has a p-value of 0.6915.
X = 0.18:
Z = -0.5
Z = -0.5 has a p-value of 0.3085.
0.6915 - 0.3085 = 0.383.
0.383 = 38.3% probability that the sample proportion is between 18% and 22%.
To learn more about the normal distribution and the central limit theorem, you can check https://brainacademy.pro/question/24663213