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Transcript
Estimating Means and
Proportions
Using Sample Means and
Proportions To Make
Inferences About
Population Parameters
Estimating Means
• Almost Always We Do Not
Know the Value of the
Population Parameters of
Interest. Hence, We Must
Estimate Their Values Based on
Sample Information.
• Estimators:
– X-bar
– P -hat
– s2
Sampling Distribution
Of The Mean
• Shape: Approximately Normal with
sufficiently large sample size (n).
• Mean: The mean of this distribution is the
same as the mean of the population from
which the sample is drawn.
• Variance: The variance of this distribution
is equal to the population variance divided
by the sample size (n).
• Justification: Central Limit Theorem.
• Implications: If we draw a random sample
of sufficient size we can estimate the mean
and variance and make probability
statements about X-bar.
Vending Machine Example
Revisited
• Recall the cup was 7oz., the
population mean and standard
deviation was 5oz. and .75oz.
respectively.
• Suppose 30 students decide to
get coffee during the break.
What is the probability that the
average cup will be less than 4
oz. ?
Vending Machine
Continued
• Step 1: P (X-bar < 4oz.)
• Step 2: Z = (4 - 5)/.75/(sq.. root
of 30)
– Z = -7.3
• Draw the picture
• P (Z < -7.3) < .001
• Interpretation: It’s very unlikely
that that the average of 30
Introduction To Point And
Interval Estimation
• Suppose we do not know the average
amount of coffee dispensed but we
do know the standard deviation
(.75oz.).
• We draw a sample of 30 students and
compute the sample mean. It turns
out to be 5.25oz.
• Point Estimate: Our best estimate of
the population mean is the sample
mean 5.25oz.
Computing An Interval
Estimate
• Suppose we want to express that
we’re not really sure what the
population mean is and would
rather put an upper and lower
bound on our estimate.
• Confidence Interval (see
formula):
Inferences About
Proportions
• Suppose that X = the number of
occurrences of a particular
event of interest (e.g., people
voting for a candidate, coin
turning up heads, people buying
a product).
• P-hat: p = x/n
• Mean of X = np
• Variance of X = npq (q = 1-p)
• Mean of p-hat = P
• Variance of p-hat = pq/n
Inferences About
Proportions Continued
• For large sample size (n) p-hat
is approximately normally
distributed (Central Limit
Theorem).
• Z = (p-hat - P)/sq. root of PQ/n
• Interval Estimate: See formula
given in class
What Proportion Of
Prospects Will Buy
• Suppose we are selling insurance
and we approach 40 prospects on a
given day. Sales records indicate that
on average about 20% of prospects
buy ( that is P = .2).
• What is the chance that the
proportion of these 40 prospects that
buy will be less than 10%?
• Suppose we were actually able to
sell 25%. What is the 95%
Confidence Interval in this situation?