Question

The amount of corn chips dispensed into a 10-ounce bag by the dispensing machine has been identified at possessing a normal distribution with a mean of 10.5 ounces and a standard deviation of 0.2 ounces (these are the population parameters). Suppose a sample of 100 bags of chips were randomly selected from this dispensing machine. Find the probability that the sample mean weight of these 100 bags is less than 10.45 ounces. (Hint: think of this in terms of a sampling distribution with sample size

Answer 1

Answer:

0.62% probability that the sample mean weight of these 100 bags is less than 10.45 ounces.

Step-by-step explanation:

To solve this question, the concepts of the normal probability distribution and the central limit theorem are important.

Normal probability distribution

Problems of normally distributed samples 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 random variable X, with mean $\mu$ and standard deviation $\sigma$, a large sample size can be approximated to a normal distribution with mean $\mu$ and standard deviation $\frac{\sigma}{\sqrt{n}}$

In this problem, we have that:

$\mu = 10.5, \sigma = 0.2, n = 100, s = \frac{0.2}{\sqrt{100}} = 0.02$

Find the probability that the sample mean weight of these 100 bags is less than 10.45 ounces

This is the pvalue of Z when X = 10.45. So

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

$Z = \frac{10.45 - 10.5}{0.02}$

$Z = -2.5$

$Z = -2.5$ has a pvalue of 0.0062.

So there is a 0.62% probability that the sample mean weight of these 100 bags is less than 10.45 ounces.