Workshop Statistics: Discovery with Data and Fathom

Topic 17: Sampling Distributions II: Means

Activity 17-1: Coin Ages (cont.)

(a) observational units: pennies;  variable: age, quantitative
(b) parameters;  mean: m;  standard deviation: q
(c) No, this distribution is skewed to the right.
Students' answers to (d)-(l) may differ since the data are chosen randomly.  These are meant to be sample answers.
(d) 13, 24, 17, 0, 0

(e) 10.8
(f)
Sample no.
1
2
3
4
5
Sample mean
10.8
13
5.8
12.8
17.8
(g) No, this is an example of sampling variability.  This is a quantitative variable, rather than a categorical one.
(h) mean of  values: 12.04;  standard deviation of x-bar values: 4.33
(i) This mean is reasonably close to the population mean.  This standard deviation is less than the population standard deviation.
(j)
(k) This distribution appears to centered near the population mean of 12.264.  The values are less spread out than the population distribution and the five sample means of size 5.
(l) yes
 

Activity 17-2: Coin Ages (cont.)


Students' answers to (a)-(n) may differ since the data is chosen randomly.  These are meant to be sample answers.
(b)

        Yes, this resembles the distribution of ages in the population.

(c)-(i)
 
 
Population mean
m = 12.264
Population std. dev.
s = 9.613
Population shape = skewed right
Sample
size
Mean of sample means
Std. dev. of sample means
Shape of sample means
1
13.76
10.314
 skewed to the right
5
 12.238
3.659 
fairly symmetric 
25
 12.392
 2.027
 normal curve
50
 12.201
 .153
steeper normal curve 

(c) The sample mean of 13.76 and the sample std.dev. of 10.314 are reasonably close to their population counterparts.  The distribution is skewed to the right, with a peak at 3 years old.
(d)

        This distribution is much more symmetric (much less skewed to the right) then either the population distribution or the distribution attained with only a sample size of 1 for each mean.  Also, this distribution has a much smaller spread.

(e) The sample mean (12.238) is very close to the actual population mean, while the sample standard deviation of 3.659 is much lower than the actual population standard deviation of 9.613.  The distribution is much more symmetric (not skewed to the right nearly as much as the population distribution).
(f)

        This distribution is more normally shaped than either the population distribution or the distributions from using sample sizes of 1 and 5.  It is also a narrowed distribution (less spread).

(g) Once again, the sample mean is close to the population mean, and the sample standard deviation is much lower than the population standard deviation.  This sample standard deviation is even lower than the sample standard deviation when using a sample size of 5.  The shape of the distribution does not resemble the population distribution at all.  This distribution is normally shaped, while the population distribution was skewed to the right.
(h)

                This distribution is much more localized than either the population distribution or the distributions from the sample sizes 1, 5, and 25.  Like the sample size of 25 (unlike the population distribution or the sample size of 1), it appears to be normally distributed, but with a very narrow spread.

(i)  The sample mean is very close to the population mean (and the sample means from the distributions using all the other sample sizes).  However, the sample standard deviation is MUCH smaller than the population standard deviation.  The distribution is much more localized around the mean (less spread out), and distributed normally as opposed to being skewed to the right.
(j)  Although answers may vary, the correct answer is yes (neither of the populations has a distribution that is close to normal).
(l)

        This appears to be a normal distribution.

(m)
 
Mean
Std. dev.
Shape
Population 1
m = .5
s= .289
uniform
Sample Means of size n = 50 from 1
.4964
.0402
 normal
Population 2
m = .5
s = .354
u-shaped
Sample means of size n = 50 from 2
.5032
.0483
 normal

(m) The sample mean is close to the population mean, while the sample standard deviation is much smaller than the population's.
(n) (l)

        This also appears to be a normal distribution.

    (m) Once again, the sample mean is close to the population mean, while the sample standard deviation is much smaller than the population standard deviation.
(o) Penny ages: 1.359;  1: .041;  2: .050; these are reasonably close to the values we obtained for the standard deviation of the 100 simulated sample means.
 

Activity 17-3: Christmas Shopping

(a)  It is a statistic because the 922 adults are meant as a sample of the population of all American adults who plan to buy Christmas gifts that year.
(b) (c) m is not necessarily equal to $857, though it is possible.  All values listed for are m possible, with the different sample mean arising by chance alone.
(d) The central limit theorem says that the sample means would vary like a normal distribution with mean $850 and standard deviation 250/sqrt(922)=8.23 if samples of size n = 922 were taken over and over.  This does not depend on the shape of the distribution of expected expenditures in the population since the sample size is large.
(e)

(f) No, since this is a normal distribution, $857 is near the center of the distribution, well within one standard deviation of the mean.
(g, d) The sample mean would vary over a normal distribution with mean $800 and standard deviation 8.23 if samples of size n = 922 were taken over and over.
(g, e) The picture would be the same, just shifted to center around 800:

(g, f) Now $857 is way in the tail of the distribution. This would be a very unlikely sample mean if the population mean was equal to $800.
(h) The CLT says that the standard deviation of the sampling distribution of the sample mean should be 250/sqrt(922) = 8.23.
(i)2(8.23)=16.46
857-16.46=840.5
857+16.46 = 873.46
interval for m: ($840.54, $873.46)