Workshop Statistics: Discovery with Data, Second Edition

Topic 17: Sampling Distributions II: Means

Activity 17-1: Coin Ages (cont.)

Click here for answers for Minitab version.
(a) observational units: pennies;  variable: age, quantitative
(b) parameters;  mean: m;  standard deviation: s
(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)-(m) may differ since the data is chosen randomly.  These are meant to be sample answers.
(a)
        Yes, it resembles the distribution of ages in the population.
(b) mean: 12.8;  standard deviation: 9.228;  Yes, they are reasonably close to their population counterparts.  The distribution is skewed to the right.
(c)
        This distribution does not have as wide a spread. The shape is much less skewed.
(d) mean: 12.426;  standard deviation: 4.072;  The mean is pretty close, but the standard deviation is lower.
(e)
        The spread is even more narrow and mound shaped now, as variability decreases.
(f) mean: 12.098;  standard deviation: 1.715;  The mean is still close, but the standard deviation is even lower now.
(g)
        Again, the distribution narrows and the variability decreases.  The distribution has become more normal.
(h) mean: 12.02;  standard deviation: 1.371;  The mean is still close, but the standard deviation is slightly lower than before.
(i) Although answers may vary, the correct answer is yes.
(j)
        This distribution is roughly normal.
(k), (l, k)
 
Mean
Std. dev.
Shape
Population 1
m = .5
s= .289
uniform
Sample Means of size n = 50 from 1
.49937
.03637
  
Population 2
m = .5
s = .354
u-shaped
Sample means of size n = 50 from 2
.50264
.04686
  
(l)
   This distribution is roughly normal.
(m) Penny ages: 1.359;  1: .041;  2: .050
 

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)