Introduction to Stats

Question

Assume that the readings at freezing on a batch of thermometers are normally distributed with a mean of 0°C and a standard deviation of 1.00°C. A single thermometer is randomly selected and tested. Find P₃₇, the 37-percentile. This is the temperature reading separating the bottom 37% from the top 63%.

P₃₇ = ___°C

Answer 1

Danya.

The area between the curve of the distribution and the x-axis will be 1.

In this problem we need to find the value X so that if we draw a vertical line from x=X upwards, the area to the left of the line is 0.37 (which means the area to the right must be 1 - 0.37 = 0.63).

If you look at a table of areas under a standard normal relating area to Z score - when the area is 0.37, the Z score is approximately -0.33 (this gives an area to the left of the vertical line as 0.3707).

Z = (X-µ) / σ. (DEFINITION). µ is the mean (here 0), and σ is the standard deviation (here 1).

Z = -0.33

So -0.33 = (X-0) / 1. -0.33 = X/1. So X = -0.33 x 1 = -0.33

So P 37 = -0.33 degrees celsius

Answer 2

Hi Danya,

You can also find the value of the P₃₇ using your graphing calculator rather than the tables. Since the distribution is normal, use the function invNorm on your TI 83/84 calculator. You can find InvNorm by pressing 2nd vars/option 3. Depending on your operating system, you will be asked for the area, the mean, µ, and the standard deviation, σ. The newer operating systems will also ask for the "Tail" and you will pick between left, center, or right. If you calculator does not ask for the tail, the area is always the area to the left.

For this problem, area: .37; µ: 0; σ: 1, Tail: Left. When you press "paste" you will see "invNorm(.37,0,1,Left)" or "invNorm(.37,0,1)." Now press enter to get the answer, which is, as Mike found, -.332 degrees. The calculator is generally easier to use than the tables and more accurate since you never have to interpolate. If you are taking AP Statistics, make sure you understand fully and completely normal distributions and how to find areas, z-scores, and observation values.

Good luck!