Download Homework 5

Homework 5
Notes:

You may use minitab whenever appropriate and include the minitab output in your
homework. To place Minitab output into your homework, highlight the portion of
the Minitab printout you want, copy that to the clipboard, and then paste it into the
document.
{Important to remember that the standard deviation of ybar is σ/sqrt(n), not σ}
1. Heights of U.S. adult females is approximately normal with a mean of 65 inches and a
standard deviation of 3.5 inches. Find the percentage of female heights that satisfy the
following:
a. Less than 58 inches.
b. Greater than 72 inches.
c. Between 58 and 72 inches.
d. Assume there are 25 U.S. adult female students in our class. What is the probability
that the mean height of these 25 students is between 58 to 72 inches?
e. The height so that 90 percent of females are less than this height.
f. A researcher in the Netherlands reads about this distribution of U.S. adult female
heights. This researcher prefers to use the metric system where one inch equals 2.54 centimeters.
In centimeters, how would expect the distribution of U.S. adult female heights be described in
terms of shape, mean and standard deviation?
2. In any given semester, there are approximately 210 students enrolled in Stat 500. They are
assigned id 1- 210. Use Minitab to generate a random sample of 10 students and list the output
in your homework. You can first generate rows of data and then sample from that by:
Calc>Make patterned data>simple set of numbers
This will bring up a dialog box -- enter the following: C1 in Store patterned data, 1 in first value,
210 in last value. You should see a list of numbers appear in column C1 from 1 to 210.
Now you want Minitab to randomly select numbers from that range by using
Calc>Random Data>Sample from column
Another dialog box will appear. Enter 10 in Sample ___ rows from columns, enter C1 in the
space directly below, and enter C2 in the “Store samples in:” space.
(You'll also see that there's an option to sample with replacement but sampling without
replacement is the default, so don't check this). Click OK. Column C2 will populate with 10
numbers randomly drawn from the series in column C1 without replacement. The results will be
displayed in columns. To save space, you can display the columns to session window by:
Data > Display Data, the data will be displayed in the session window. You can then copy and
paste those data to your homework.
3. The weight limit for an elevator in your building is set at 2200 pounds. Assume the weight of
typical adult is a random variable with a mean of 180 pounds and a standard deviation of 12
pounds. If the elevator has 13 adults, what is the probability that the total weight of that elevator
will exceed the 2200-pound limit? [hint: first formulate the sample total problem to a sample
mean problem.]
4. According to the rules of tennis, a tennis ball is supposed to weigh between 56.7 grams and
58.5 grams. A tennis ball manufacturing machine produces balls with a mean of 57.6 grams with
a 0.3 grams standard deviation. Assume that this distribution of weights is approximately
normal.
a. If the machine is working properly, find the probability that a ball manufactured by this
machine satisfies the rules.
b. After the machine has been used for a year, the process still has a mean of 57.6 grams but due
to machine wear the standard deviation is now 0.6 grams. Find the probability that a
manufactured ball satisfies the rules.
c. For the properly operating machine, describe the sample mean distribution for the production
of 16 tennis balls. Include the distribution name, mean and standard deviation of this
distribution.
5. Simulation for sampling distribution
Read lesson 5.2 and then go to http://onlinestatbook.com/rvls.html
>Simulation/Demonstration
>sampling distribution simulation
Read the instruction and then choose begin
a. Use the population distribution as the default one: normal distribution
Choose the statistics “mean” for the bottom two graphs.
i)
ii)
Use sample size N = 2 for the next to bottom graph
Use N = 5 and then N = 25 for the bottom graph.
For simulation purpose, you can start with animated samples to see how things work. Then to
get results to fill in the following tables, you will choose 10,000 samples (this will get
samples 10,000 times. Each time, sample of size N is taken and the statistics for the sample
computed) so that it will simulate the sampling distribution well.
Fill in the blanks below:
population mean = ____, population st. dev. = ___
sample size
mean of sample
mean
st. dev. of sample
mean
Skew
N=2
N=5
N=25
b. Change the population to skewed. Follow the same procedure as in a.
population mean = ___, population st. dev. = ___
sample size
mean of sample
mean
st. dev. of sample
mean
skew
N=2
N=5
N=25
c. State the central limit theorem. Do the simulation results indicate that the central limit
theorem seem correct? Explain in terms of the mean, the standard deviation, the skew and the
distribution. (Note that the results are from simulation, so the skew may differ slightly from
0 due to simulation error even though the distribution is normal.)