MONT 105Q – Mathematical Journeys Discussion – Using the Standard Normal Table March 21, 2016

Background

Right-hand tail probabilities for a standard normal random variable (i.e. normal distribution with μ = 0, σ^{2} = 1) are given in the table on the attached sheet. For instance, The entry for z = 1.43 is .0764. This means that

P (Z > 1.43) = .0764.

From this we could also derive facts such as

P (Z ≤ 1.43) = 1 − .0764 = .9236.

Moreover, by the symmetry of the standard normal curve, if c > 0, we can see

P (0 < Z < c) = \frac{1}{2} − P (Z > c) = P (−c < Z < 0).

Hence, for instance,

P (−1.0 < Z < 2.0) = P (−1 < Z ≤ 0) + P (0 < Z < 2)

= (\frac{1}{2} − P (Z > 1)) + (\frac{1}{2} − P (Z > 2))

= (.5 − .1587) + (.5 − .0228) = .3413 + .4772 = .8185.

For today’s discussion problems, the following key fact will be needed too.

Key Fact: If Y is normal with mean μ and standard deviation σ, then

Z = \frac{Y − μ}{σ}

is standard normal, and the table can be applied to Z.

For Example: Say we know that Y has a normal distribution with mean μ = 12 and SD σ = 2. Suppose we want to know the probability P (Y > 11.3). We can determine this from the Key Fact as follows:

Y > 11.3 ⇔ Z = \frac{Y − 12}{2} > \frac{11.3 − 12}{2} = −.35, so
P (Y > 11.3) = P (Z > −.35)

= 1 − P (Z > +.35) by symmetry = 1 − .3632 from table = .6368 In today’s discussion, you will practice using the standard normal table to answer questions about normally distributed quantities following similar methods.

Discussion Questions

A) Let Z be a standard normal.

1) Find P (Z > 1.29).

2) Find P (Z < 1.0).

3) Find P (−2.13 < Z < −0.56).

4) Find c such that P (Z > c) = .05. (You’ll need to “interpolate” here. Do you see why?)

B) Y is normally distributed with mean 6 and SD 4. Find

1) P (Y < 7).

2) P (Y > 8.5).

3) P (5 < Y ≤ 8).

C) SlimMints are sold in two-packs with a stated label weight of 20.4 grams. The actual weights of the packages are normally distributed with mean μ = 21.37 and SD σ = .4.

1) Let Y be the weight of a single package selected at random from the production line. What is the probability P (Y > 22.07)?

2) Suppose that 15 packages are selected independently. Let X be the number that weigh less than 21 grams. What is P (X = 2) = the probability that exactly two have weight less than 21 grams?

We’ll compare notes and I’ll give the correct answers at the end of the class.