Question

An industrial process produces batches of a chemical whose impurity levels follwo a normal distribution with standard deviation 1.6 grams of chemical. A random sample of 100 batches is selected in order to estimate the population mean impurity level. The probability is 0,0367 that the sample mean impurity level exceeds the population mean by how much?

​

Answer 1

The probability is 0,0367 that the sample mean impurity level exceeds the population mean by 0.2864 grams of chemical.

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 z-score of a measure X is given by:

`Z = (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 p-value, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem establishes 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 = (\sigma)/(√(n))`.

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

Standard deviation 1.6 grams of chemical. Random sample of 100.

This means that
`n = 100, s = (1.6)/(√(100)) = 0.16`

The probability is 0,0367 that the sample mean impurity level exceeds the population mean by how much?

​Z multiplied by s, in which Z has a p-value of 1 - 0.0367 = 0.9633, so Z = 1.79.

1.79*0.16 = 0.2864.

The probability is 0,0367 that the sample mean impurity level exceeds the population mean by 0.2864 grams of chemical.