The amount of a certain chemical in a type A cell is normally distributed with a mean of 10 and a standard deviation of 1. The amount in a type B cell is normally distributed with a mean of 14 and a standard deviation of 2. To determine whether a cell is type A or type B, the amount of chemical in the cell is measured. The cell is classified as type A if the amount is less than a specified value c, and as type B otherwise.

a-If c=12, calculate the probability that a type A cell will be misclassified, and the probability that a type B cell will be misclassified.

b-Find the value of c for which the two probabilities of misclassification are equal.

Hey! Have you tried drawing the two distributions on a graph to visualise the problem at all? I think that would be a good start. Here's the diagram I pulled up to help myself with this:

Draw yourself the two distributions for Type A and Type B, draw a dotted line at 12, and hopefully these will guide you:

- Consider just Type A for a bit. It is mean 10, std-dev of 1. 12 is two standard deviations above the mean. This means that only about 2.2% of Type A cells will have more than 12 units of this chemical. So there is a 2.2% chance that a Type A cell is misclassified.

- For Type B, it has a larger standard deviation (there is more spread in the distribution). If you had a to scale drawing provided, you'd see that the Type B curve spreads below the line at c=12 more than the Type A one spreads above it. 12 is only standard deviation below the mean of Type B cells. Therefore, you see a 15.8% chance of being misclassified.

For the last bit, you need to find where to draw the line such that the percentage of Type A cells above the line, is equal to the percentage of Type B cells below the line. Conceptually, you are looking for the value which is equally 'far' from the centre of the two distributions, when you consider that one standard deviation is larger than the other. Do you know of a numerical measure which takes into account mean and standard deviation in this way?

Have a think and if you're stuck...

CLICK ME IF STUCK

Use z scores! The misclassification rate will be the same when the z-score is the same (except one will be positive, one will be negative, because \(c\) sits on opposite sides of the mean for each distribution).

Solve for your answer

Hopefully this helps!