Sampling distributions
Intros
Examples
Lessons
- Understanding Proportions and Sample Size
There are 7 supercars competing on a racetrack. The cars are labelled {1, 2, 3, 4, 5, 6, 7}.
Find, - Connecting Sample Proportions to Population Proportions
Alice, Bob, Cole, Daisy, and Eve are all the students in a 10th grade class. Alice, Cole and Eve brought a lunch to class while the rest of the class did not. - Connecting Sample Means to the Population Mean
Uncle Sammy grows prize winning big zucchinis. The weights of his four largest zucchinis are given in the table below:
Zucchini:
A
B
C
D
Weight:
7 lbs
9 lbs
11 lbs
5 lbs
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Topic Notes
Sampling distributions
A sampling distribution is a probability distribution of a statistic obtained through a large number of samples drawn from a specific population. The sampling distribution of a given population is the distribution of frequencies of a range of different outcomes that could possibly occur for a statistic of a population.Sample proportion will estimate population proportion
Sample mean will estimate population mean
$\quad N = population \;size$
$\quad n = sample \;size$
$\quad \mu = population \;mean$
$\quad \overline{x} = sample \;mean$
$\quad p = population \;proportion$
$\quad \hat{p} = sample\; proportion$
Sampling distribution model
Example 1
Understanding Proportions and Sample SizeThere are 7 supercars competing on a racetrack. The cars are labelled {1, 2, 3, 4, 5, 6, 7}.
Find:
And so, there are a total of 343 possible different samples of the 7-car population when it is allowed to repeat cars within the sample (with replacement).
Therefore, in this case we have that:
We will talk about combinations extensively in later lessons, for now, you can think of it as the method to obtain the possible amount of different samples that can be obtained from a population when there is no replacement (which happens to be the most logical scenario in many cases), just as it has been defined with this problem.
Example 2
Alice, Bob, Cole, Daisy, and Eve are all the students in a 10th grade class. Alice, Cole and Eve brought a lunch to class while the rest of the class did not.Notice that this is because three people from the class brought lunch, out of a total of 5. So our total population was 5 people, and the sample who brought lunch was 3.
And so, there are 10 possible samples of two people at a time coming from the 10th grade class forming the next list:
With that in mind, since there are a total of 10 possible different pair samples, then each of those pairs has a 1/10 of probability of occurring (or being selected).
Combination |
P(x) = probability |
$\hat{p}$ = sample proportion |
{Alice, Bob} |
1/10 |
0.5 |
{Alice, Cole} |
1/10 |
1 |
{Alice, Daisy} |
1/10 |
0.5 |
{Alice, Eve} |
1/10 |
1 |
{Bob, Cole} |
1/10 |
0.5 |
{Bob, Daisy} |
1/10 |
0 |
{Bob, Eve} |
1/10 |
0.5 |
{Cole, Daisy} |
1/10 |
0.5 |
{Cole, Eve} |
1/10 |
1 |
{Daisy, Eve} |
1/10 |
0.5 |
As you can see, the average of the sample proportions for the people who brought lunch to school is the same as the proportion of the population that brought lunch. Therefore, in here we reaffirm what we mentioned at the beginning of this lesson: Sample proportion will estimate population proportion.
Example 3
Uncle Sammy grows prize winning big zucchinis. The weights of his four largest zucchinis are given in the table below:And so the average weight of uncle Sammys zucchinis is 8lbs.
And so, there are 6 different possible samples of two zucchinis from the total of uncle Sammys ones and they conform the next list:
Remember, the average weight of each sample is what we call the sample mean.
Combination |
P(x) = probability |
$\overline{x}$ = Avg. weight |
{A, B} |
1/6 |
$\overline{x} = \frac{7+9}{2} =8 lbs$ |
{A, C} |
1/6 |
$\overline{x} = \frac{7+11}{2} = 9 lbs$ |
{A, D} |
1/6 |
$\overline{x} = \frac{7+5}{2} = 6 lbs$ |
{B, C} |
1/6 |
$\overline{x} = \frac{9+11}{2} = 10 lbs$ |
{B, D} |
1/6 |
$\overline{x} = \frac{9+5}{2} = 7 lbs$ |
{C, D} |
1/6 |
$\overline{x} = \frac{11+5}{2} = 8 lbs$ |
And so, we reaffirm what was mentioned at the beginning of this lesson: Sample mean will estimate population mean.
To finalize this lesson here are a few recommendations to you: A lot of terminology is covered on this lesson for the sampling distribution of the sample mean, along with a simple example. Then, this page on the sample proportion provides a review on the formulas we have used during this lesson. Both links can be useful to you to reaffirm what was learnt today and to aid in your independent studies.
This is it for our lesson of today, see you in the next one!
Basic Concepts
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