Download Exercises: Distribution of the Sample Mean

Exercises: Distribution of the Sample Mean
1. Women’s heights are Normally distributed with a mean  = 63.5 inches and a standard
deviation  = 2.5 inches.
a) If 1 woman is randomly selected, what is the probability that her height is between 63.0
and 64.0 inches?
Suppose instead 9 women are randomly selected.
b) What is the distribution of the (sample) mean height for these 9 women? State the shape,
mean and standard deviation. Interpret the values of the mean and standard deviation.
c) What is the probability the (sample) mean height is between 63.0 and 64.0 inches.
d) Using a relative frequency interpretation, interpret this probability by filling in the
blanks: 45.15% of all _______ [units] have _____ [variable] between 63.0 and 64.0
inches.
e) Answer b and c if instead 36 women are randomly selected.
f) Complete the table.
Probability between 63.0 and 64.0
One woman’s height
Mean of 9 women’s
heights.
Mean of 36 women’s
heights.
g) Without doing any computing, explain what happens to…
…the distribution of the (sample) mean as the sample size increases.
…the probability that the mean for the sampled women is between 63.0 and 64.0 as the
sample size increases.
2. Engineers must consider the breadths of male heads when designing helmets. Men have head
breaths that are Normally distributed with a mean of 6.0 inches and a standard deviation of
1.0 inch.
a) If one male is selected at random, find the probability that his head breadth is less than
6.2 inches.
Suppose instead 10 men are randomly selected.
b) What is the distribution of the (sample) mean breadth for these 10 men?
c) If 10 men are randomly selected, what is the probability that they have a mean breadth
less than 6.2 inches?
d) Answer b and c if instead 100 men are randomly selected.
e) Complete the table.
Probability below 6.2
Breadth of one man’s head
Mean breadth of 10 men’s heads
Mean breadth of 100 men’s heads
f) If the sample size is increased, what happens to…
…the distribution of the (sample) mean
…the probability that the mean for the sampled men is less than 6.2 inches.
3. Your company uses varying amounts of a chemical compound each month. The amounts
vary according to a Normal distribution with mean 150 kg and standard deviation 16 kg.
a) Find the probability more than 160 kg is required in May.
b) Consider a sample of 6 months. What is the distribution for the (sample) mean amount of
chemical used?
c) Find the probability the mean monthly usage for May – October (6 months) is more than
160 kg.
d) Find the probability the mean monthly usage for an entire year is more than 160 kg.
e) Complete the table.
Probability more than 160
One month
Six months
Twelve months
f) Explain what happens to the probability that the mean is more than 160 as the sample size
increases.
g) What is the probability the total usage for an entire year is more than 1920 kg? (Hint:
Rephrase this as a question about the mean usage.)
4. Weights of German Shepherd puppies are Normally distributed with mean 17 ounces. Which
of these is most likely? Least likely?
a single puppy over 18
ounces
8 puppies with mean over 18 24 puppies with mean over
ounces
18 ounces
5. A car circles a 2.5 miles track in lap times that are independent and vary according to a
Normal distribution with a mean of 59 seconds.
a) On average, about how fast is the car going (in mph to the nearest 0.01 mph)?
b) Which is most likely? Which is least likely?
One lap in under 1 minute.
Two laps in under 2
minutes.
Three laps in under 3
minutes.
(What would the average time per lap be for each choice? Now think about standard
deviations as the sample size increases.)
Solutions
1.
a) 0.1585.
b) Normal with mean 63.5 and standard deviation 0.8333. The sample mean for all possible
samples of 9 women varies according to a Normal distribution with mean 63.5 inches and
standard deviation 0.8333 inches.
c) 0.4515.
d) For 0.4515 (= 45.15%) of all possible samples of 9 women, the sample mean is between
63.0 and 64.0 inches. “samples of 9 women” are the units; “sample mean” is the variable.
e) 0.7699.
f) See the answers to a, c and e.
g) It is always Normal; the mean is always 63.5; but the variation (standard deviation)
shrinks, making the distribution more and more concentrated over 63.5 – eventually
becoming a spike. The probability will get larger and larger (because 63.5 – which is where
the spike would be – is between 63.0 and 64.0).
63
64
Variable
n =1
n =9
n = 36
56.0
2.
58.5
61.0
63.5
66.0
68.5
71.0
a) 0.5793.
b) Normal with mean 6.0 and standard deviation 0.3162.
c) 0.7365.
d) It is Normal with mean 6.0 and standard deviation 0.1. The probability is now 0.9772.
e) See the answers to a, c and d.
f) It is always Normal; the mean is always 6.0; but the variation (standard deviation)
shrinks, making the distribution more and more concentrated over 6.0 – eventually
becoming a spike. The probability of a result below 6.2 will get larger and larger (because
6.0 – which is where the spike will be – is below 6.2).
3.
a) 0.2660.
b) Normal with mean 150 and standard deviation 6.532.
c) 0.0629.
d) The standard deviation is now 4.619. The probability is 0.0152.
e) See the answers to a, c and d.
f) The probability decreases. Because the distribution is getting closer to being a spike over
150, the probability of a result above 160 gets smaller and smaller.
g) 0.0152. (If the total for 12 months is above 1920, the sample mean is above 1920/12 =
160. This was answered in part d.)
160
Variable
n = 12
n =6
n =1
102
114
126
138
150
162
174
186
198
4.* A single puppy over 18 is most likely. 24 puppies with mean over 18 is least likely. As the
sample size grows the distribution of the sample mean approaches a spike over 17. Results
above 18 get less and less likely.
5.
a) mph = 2.5 miles  (59 seconds)  60 seconds  1 minute  60 minutes  1 hour = 152.54
mph.
b) All these statements convert to “sample mean lap time less than 60 seconds.” The
distribution of the sample mean approaches a spike over 59 as the sample size increases. So
this probability will get larger and large as the number of laps increases. One lap in under 1
minute is least likely. Three laps in under 3 minutes is most likely.
*
You could solve both 4 and 5 by making up a value for the standard deviation. See what happens. Change the
value and see whether the same thing happens. Convince yourself that the value of the standard deviation is not
important to the qualitative nature of the comparisons that you’re being asked to make.