Section 7.1 – Sampling Distributions
Inferential Statistics vs. Descriptive Statistics
Inferential: RANDOMIZATION makes it possible to use probability to make inferences (the whole point of
statistics – to make decisions)
Descriptive: Describing a set of data. (What we’ve done so far)
Sample
µ
σ
p
x
Statistic
*Fixed values, but
often unknown
Parameter
Population
s
p̂
*values vary from sample
to sample
*used to estimate an
unknown parameter
*KNOWN – calculated
from data
Sampling Variability
The value of a statistic varies in repeated random sampling.
Example, p. 425: Identify the population, the parameter, the sample, and the statistic in each of the following
settings:
(a) The Gallup Poll asked a random sample of 515 U.S. adults whether or not they believe in ghosts. Of the
respondents, 160 said “Yes”.
160
Population: U.S. adults
Parameter: p
Sample: 515 U.S. adults
Statistic: 𝑝̂ = 515 = 0.31
(b) During the winter months, the temperatures outside the Starnes’s cabin in Colorado can stay well below
freezing for weeks at a time. To prevent the pipes from freezing, Mrs. Starnes sets the thermostat at
50°F. She wants to know how low the temperature actually gets in the cabin. A digital thermometer
records the indoor temperature at 20 randomly chosen times during a given day. The minimum reading
is 38°F.
Population: All times during the day
Parameter: µ
Sample: 20 temperatures at randomly selected times
Statistic: sample minimum = 38°F
Check Your Understanding, p. 425: Label the bold faced values as either a parameter or a statistic.
1. On Tuesday, the bottles of Arizona Iced Tea filled in a plant were supposed to contain an average of 20
ounces of iced tea. Quality control inspectors sampled 50 bottles at random from the day’s production.
These bottles contained an average of 19.6 ounces.
Parameter: µ = 20 ounces
Statistic: 𝑥̅ = 19.6 ounces
2. On a New York-to-Denver flight, 8% of the 125 passengers were selected for random security screening
before boarding. According to the Transportation Security Administration, 10% of passengers at this
airport are chosen for random screening.
Parameter: p = 0.10
Statistic: 𝑝̂ = 0.08
***What happens if we took many samples?***
- Take a large number of samples from the same population
- Calculate sample mean x or sample proportion p̂ for each sample
- Make a graph of the values x or p̂
- Examine the distribution displayed (SOCS)
Could be expensive!
We can use simulation.
Example: Everyone in class will use their calculator to find a random sample of 10 numbers between 1 - 10.
# 𝑜𝑓 3′𝑠 𝑜𝑟 𝑏𝑒𝑙𝑜𝑤
Count how many are less than or equal to 3. Find the proportion that are 3 or below =
. Collect
10
every student’s proportion and create a histogram. Then, describe the histogram. Below is an example:
N = 50 trials
𝑝̂
0
0.1
0.2
0.3
0.4
0.5
0.6
Count
3
9
14
10
9
3
2
Shape: Roughly symmetrical
Outliers: No apparent outliers
Center: Between 0.2 and 0.3
Spread: 0 to 0.6