Download 1. A sample of size 40 is taken from a much larger population with

Answers to In-Class Exercises 16+2
1. A sample of size 40 is taken from a much larger population with mean 20 and standard deviation 3.
(a) Can the CLT be used to determine µX̄ and σX̄ ? Why or why not?
Yes: the sample size is large enough.
(b) If your answer to part (a) was Yes, find µX̄ and σX̄ .
µX̄ = µx̄ = 20
σX̄ = √340 ≈ 0.4743
2. A sample of size 28 is taken from a much larger population with mean 20 and standard deviation 3.
(a) Can the CLT be used to determine µX̄ and σX̄ ? Why or why not?
No: the sample size is too small for a distribution not known to be normal.
(b) If your answer to part (a) was Yes, find µX̄ and σX̄ .
µX̄ =
σX̄ =
3. A sample of size 28 is taken from an approximately normally distributed population with mean 20 and
standard deviation 3.
(a) Can the CLT be used to determineµX̄ and σX̄ ? Why or why not?
Yes: although the sample size is small, the population is known to be approximately normally distributed.
(b) If your answer to part (a) was Yes, find µX̄ and σX̄ .
µX̄ = 20
σX̄ = √328 ≈ 0.5669
4. Samples of size 36 were drawn from a much larger, normally distributed data set with mean 100 and standard
deviation 8. Let X be a random variable representing the data. Sketch the normal curves for each of X and
X together on the axis below. Be sure to label the two curves.
You already know the mean and standard deviation for the X curve, and the CLT tells you how to get the
mean and standard deviation for the X̄ curve. For the sketch, remember that the inflection points are above the
x-values (mean − std dev, mean + std dev):
5. A widget-making machine turns out widgets with framises averaging 0.82 metric inches and with standard
deviation 0.13 metric inches. The framis measurements are approximately normally distributed.
(a) If a widget is chosen at random, what is the probability that its framis is exactly one metric inch long?
C’mon, how many times have I asked this now? The probability of choosing any one point under a continuous
pdf is 0.
(b) If a widget is chosen at random, what is the probability that its framis is at least one metric inch long?
Let X = framis length. Then P(X > 1) = normalcdf(1, 10 000, 0.82, 0.13) ≈ 0.08309
(c) A sample of 100 widgets had a mean framis length of 0.88 metric inches. Would you say that the machine
is working correctly? Back up your conclusion with a calculation.
You want to find the probabilty of seeing a sample mean at least as far away from the population mean as the
one observed if the machine is working correctly. Note that the observed sample mean, 0.88, is larger than the
(supposed) population mean, so “at least as far away from the population mean as 0.88” means “0.88 or more.”
So, what we do here is to assume that µ = 0.82 and σ = 0.13, and find P(X̄ ≥ 0.88). We have σx̄ = √0.13 =
100
0.013, so
P(X̄ > 0.88) = normalcdf(0.88, 10 000, 0.82, 0.013) ≈ 1.9639 × 10−6 = 0.00000196
That’s a really tiny probability. In fact, if the machine is working correctly, you should see a sample mean of
0.88 or more only about two times out of a million. Thus, the fact that you did see a sample mean of 0.88
means that it’s very unlikely that the machine is working correctly.
–2–