HW # 4 – Sampling Distributions
1. (IN CLASS) A shipment of ball bearings has a mean diameter of 2.5003 cm. This is within the
specifications for acceptance of the shipment by the purchaser. By chance an inspector chooses
100 bearings from the lot that have a mean diameter of 2.5009 cm. Because this is outside the
specified limits the lot is mistakenly rejected. For each boldface number tell if it is a statistic or a
parameter.
2. (ANSWER GIVEN) A shipment of ball bearings has a mean diameter of 2.5003 cm. By
chance an inspector chooses 100 bearing from the lot that have a mean diameter of 2.5009 cm.
We would actually like to know the mean diameter of all ball bearings made by this company.
For each boldface number tell if it is a statistic or a parameter.
3. (HOMEWORK) Voter registration records show that 68% of all voters in Indianapolis are
registered as Republicans. To test a random digit dialing device, you use the device to call 150
randomly chosen residential telephones in Indianapolis. Of the registered voters contacted 73%
are registered Republicans. For each boldface number tell if it is a statistic or a parameter.
4. (ALTERNATE HW) How does caffeine affect our bodies? In a matched pairs experiment,
subjects pushed a button as quickly as they could after taking a caffeine pill and also after taking
a placebo. The mean pushes per minute were 283 for the placebo and 311 for the caffeine. For
each boldface number tell if it is a statistic or a parameter.
5. Let’s illustrate the idea of a sampling distribution in the case of a very small sample from a
very small population. The population is the scores of 10 students on an exam:
Student 0 1 2 3 4 5 6 7 8 9
Score
82 62 80 58 72 73 65 66 74 62
Find the population mean. Also start anywhere you want in the table of random digits and pick
10 random samples of the appropriate size. Make a histogram of your 10 values with intervals
60-62, 62-64, etc.
A) (IN CLASS) Do this for samples of size 5.
B) (SOLUTION GIVEN) Do this for samples of size 3.
C) (HOMEWORK) Do this for samples of size 4.
D) (ALTERNATE HW) Do this for samples of size 6.
6. (IN CLASS) Most people think that the average body temperature in adult humans is 98.6.
However, this figure is based on data from the 1800’s. In a 1992 article in the Journal of the
American Medical Association, it is reported a more accurate figure is 98.2. Assume a
normal model is appropriate and that the standard deviation is 0.7.
A) If one person is picked at random what is the probability their temperature will exceed 98.6?
B) If 8 people are picked at random what is the probability their average temperature will exceed
98.6?
C) If 12 people are picked at random what is the probability their average temperature will be below
98?
D) For all possible groups of 25 people what range of average temperatures make up the middle 95%?
E) What is the average temperature for the coolest 10% of groups of 25 people?
F) What is the average temperature for the hottest 20% of groups of 25 people?
7. (ANSWER GIVEN) Suppose the measurements on the stress needed to break a type of bolt
follow a Normal distribution with a mean of 75 kilopounds per square inch(ksi) and a
standard deviation of 8.3 ksi.
A) If one bolt is picked at random what is the probability its breaking strength will exceed 80ksi?
B) If 7 bolts are picked at random what is the probability their average breaking strength will exceed
80ksi?
C) If 9 bolts are picked at random what is the probability their average breaking strength will be below
72ksi?
D) For all possible groups of 36 bolts what range of breaking strengths make up the middle 95%?
E) What is the average breaking strength for the strongest 10% of groups of 36 bolts?
F) What is the average breaking strength for the weakest 25% of groups of 36 bolts?
8. (SOLUTION GIVEN) Assume the cholesterol levels of adult American women can be described by
a Normal model with a mean of 188 mg/dL and standard deviation of 24.
A) If one woman is picked at random what is the probability her cholesterol will exceed 200?
B) If 6 women are picked at random what is the probability their average cholesterol will exceed 200?
C) If 15 women are picked at random what is the probability their average cholesterol will be below
190?
D) For all possible groups of 16 women what range of average cholesterol levels make up the middle
95%?
E) What is the average cholesterol level for the highest 10% of groups of 16 women?
F) What is the average cholesterol level for the lowest 5% of groups of 16 women?
9. (HOMEWORK) Biological measurements on the same species often follow a Normal distribution
quite closely. The weights of seeds of a variety of winged bean are approximately Normal with a
mean of 525 mg and a standard deviation of 110 mg.
A) If one seed is picked at random what is the probability its weight will exceed 500?
B) If 11 seeds are picked at random what is the probability their average weight will exceed 500?
C) If 14 seeds are picked at random what is the probability their average weight will be below 510?
D) For all possible groups of 64 seeds what range of average seed weights make up the middle 95%?
E) What is the average weight for the heaviest 1% of groups of 64 seeds?
F) What is the average weight for the lightest 20% of groups of 25 seeds?
10. (ALTERNATE HW) Suppose the heights of women aged 20-29 follow approximately a
Normal distribution with a mean of 64 inches and a standard deviation of 2.7 inches.
A) If one woman is picked at random what is the probability her height will exceed 69.3 inches?
B) If 3 women are picked at random what is the probability their average height will exceed 69.3
inches?
C) If 16 women are picked at random what is the probability their average height will be below 62
inches?
D) For all possible groups of 81 women what range of average cholesterol levels make up the middle
95%?
E) What is the average height for the tallest 15% of groups of 81 women?
F) What is the average height for the shortest 5% of groups of 81 women?