Question

Betty's bite-size candies are packaged in bags.

The number of candies per bag is normally distributed, with a mean of $50$ candies and a standard deviation of $3$.

At a quality control checkpoint, a sample of bags is checked, and $4$ bags contain fewer than $47$ candies.

How many bags were probably taken as samples?

Answer 1

Step-1: Find the proportion of bags that have fewer than $47$ candies:

Recall the Empirical Rule that in a normally distributed population, the proportion of elements that have a value lesser than one standard deviation below the mean is $\approx 16\%$.

Since the number of candies per bag is given to be normally distributed with a mean of $50$ candies and a standard deviation of $3$, about $16\%$ of the bags have fewer than $50-7=47$ candies.

Step-2: Find the sample size that will include $4$ bags having fewer than $47$ candies:

It is given that $4$ bags are found to be having fewer than $47$ candies.

This number is expected to be $16\%$ of the size of the sample $N$.

Express this as an equation: $4=16\%\text{of}N$.

Solve this equation to get the expected sample size $N$:

$\begin{array}{rrcl}& 4& =& 16\%\text{of}\mathrm{N}\\ \Rightarrow & 4& =& \frac{16}{100}\times \mathrm{N}\\ \Rightarrow & \frac{4\times 100}{16}& =& \mathrm{N}\\ \Rightarrow & 25& =& \mathrm{N}\end{array}$

Thus, the expected sample size is $\mathrm{N}=25$.

Hence, it is expected that $\mathbf{25}$ bags were taken as a sample.