Consider exercise 6.38 on page 399 of your textbook.

Plastic bags used for packaging produce are manufactured so that the breaking
strength of the bag is normally distributed with a mean of 5 pounds per square
inch and a standard deviation of 1.5 pounds per square inch. A sample of
25 bags is selected.

So we have that the breaking strength of the bags is normal
with
lbs/*in*^{2} and
lbs/*in*^{2}.
Also, n = 25.

(a)(1) What is the probability that the average breaking strength is between
5 and 5.5 pounds per square inch?

`normalcdf`

command, we actually have to plug in
and
.
You should always draw a curve, and give it some thought.
Use your common sense to decide if the
answer the calculator comes up with is reasonable.
(a)(2) What is the probability that the average breaking strength is between
4.2 and 4.5 pounds per square inch?

(a)(3) What is the probability that the average breaking strength is less
than 4.6 pounds per square inch?

(b) Between what two values symmetrically distributed around the mean
will 95% of the average breaking strengths be?

The values asked for here will be the 2.5^{th} percentile,
and the 97.5^{th} percentile, since 95%
of the average breaking strengths will be between that.

We need the Z quantile corresponding to the .025 tail area.

So the
value, once we "un-standardize", is:

(c) What will your answers be to (a) and (b) if the standard deviation
is 1.0 pound per square inch?

Do this part on your own!

Here are the answers:

(a)(1) .49379
(a)(2) .00617799
(a)(3) .002275
(b) (4.608, 5.392)

Consider exercise 6.46 on page 402-403 of your textbook.

Historically, 93% of the deliveries of an overnight mail service arrive
before 10:30 the following morning. Random samples of 500 deliveries
are selected.

So we have a "historic" (read: population) proportion of p = .93.
Also, n = 500.

(a) What proportion of the samples will have between 93% and 95% of the
deliveries arriving before 10:30 the following morning?

`normalcdf`

command, we actually have to plug in p =.93 and
.
(b) What proportion of the samples will have more than 95% of the
deliveries arriving before 10:30 the following morning?

(c) If samples of size 1000 are selected, what will your answers be in
(a) and (b)?

Do this part on your own!

Here are the answers:

(a) .49340852
(b) .00659148

(d) Which is more likely to occur - more than 95% of the deliveries
in a sample of 500 or less than 90
before 10:30 the following morning?

We will answer the question of which is more "likely" by simply
calculating the probability of each event:

Therefore, more than 95% of the deliveries in a sample of 500 arriving before 10:30 the following morning is more likely to occur, since it has a higher probability.