STAT 1350 Lecture Notes - Lecture 17: Simple Random Sample, Pumpkin Pie Spice, Statistic
27 Apr 2018
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3/8/2018
Chapter 18 and 21: More on Probability Models/Introduction to Confidence Intervals
● Review
○ Remember the claim that 50% of Reese’s Pieces are orange. Based on what we
know about sampling distributions, which of the following is most likely (or has
the highest probability), and why?
○ A.Obtaining a random sample of 50 Reese’s Pieces with 20% or fewer orange.
○ B.Obtaining a random sample of 50 Reese’s Pieces with 60% or more orange.
● More Review
○ The proportion of orange M&Ms in the population is 0.20. What do you think is
LEAST likely to be observed?
■ A. The proportion of orange M&Ms in a random sample of 30 candies will
be greater than .30.
■ B. The proportion of orange M&Ms in a random sample of 100 candies
will be greater than .30.
■ C. Both A and B are equally likely to happen.
● Review: Probability Models for Random Samples
○ Choosing a simple random sample from a population and computing a statistic
from it is a random phenomenon.
■ 1. There are several possible outcomes (values that the statistic could take
on if different samples are chosen).
■ 2. If we use a random procedure to select our sample, we don’t know
ahead of time which sample we will select and so we also don’t know the
value of the statistic ahead of time.
■ 3. As you might recall, there is a regular distribution of outcomes if we
repeat our procedure many times.
● More on Probability Models for Random Samples
○ The distribution of a statistic tells what values the statistic takes on and how often
it would take that value if we took a lot of samples.
○ The distribution of a statistic is called a sampling distribution.
○ The sampling distributions of proportions and averages (or means) have a familiar
pattern: they can be closely approximated by the Normal curve under appropriate
conditions.
● More on the Sampling Distribution of the Sample Proportion
○ If the sample size is large enough, we know that:
■ The sampling distribution is approximately Normal in shape.
■ The center of the sampling distribution is equal to the population
parameter (in this case, the population proportion or p).
■ The standard deviation of the sampling distribution can be determined by
solving this equation:
● The square root of p times (1 minus p) over n
● What do you think?
○ It has been reported that 34% of Americans think that pumpkin spice is the best
flavor associated with fall. Imagine that we take many random samples of n =
100 from this population. Our resulting sampling distribution will be centered at a
mean of 0.34, with a standard deviation of ______.
● What is large enough?
○ If you want to determine if you have a “large enough” sample size to assume an
approximately Normal sampling distribution of the proportion, the following two
things should hold:
○ (where n = sample size and p = population proportion)
○ If you do not know p but do know the sample proportion ( ), the following should
hold:
● Confidence Intervals
○ •By constructing a confidence interval, we can take a sample statistic and use it to
estimate an unknown population parameter.
○ Remember when we did this back in Chapter 3? We estimated the margin of
error and then used it to construct a confidence statement.
○ IMPORTANT: FORGET about that “quick estimate” for the margin of
error!
○ •Now, we’ll learn a better way to compute a margin of error.
