Download Sampling distribution of the mean

BI230 Biostatistics
Chapter 5
Continuous distributions: uniform,
normal, exponential
Probability density function
•  p : R → [0,1] such that
–  for all x, 0 ≤ p(x) ≤ 1
– 
•  P( {x: a ≤ x ≤ b}) is definite integral of
density function between a and b
Expectation, stdev of
continuous r.v. X
Uniform density function
If A=-1 and B=+1, then uniform density curve is below:
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Normal density function
•  symmetric about mean µ
•  Z-score = (x-µ)/σ
•  p(x) equals a constant times
Standard normal density
•  In standard normal
–  µ = 0
–  σ = 1
Standard normal density function
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By symmetry about x-axis, the mean of
standard normal distribution equals 0
Integration by parts shows variance of
standard normal equals 1
Cumulative distribution for
standard normal
This expression equals probability that X ≤ x
P( {x: -infinity < u ≤ x} )
Integral cannot be solved analytically.
Use Table A-2 at back of book, or use
NORMDIST(x,0,1,TRUE) in Excel
Applications of normal
distribution
Continuity correction
•  Continuity correction used when the
continuous normal distribution used to
approximate the discrete binomial
distribution.
Use Φ to approximate binomial
probability in following examples
Suppose p=0.1, n=100, so µ = 10,
and σ=sqrt(npq)=3
•  P(less than 30 successes) = P(k < 30)
•  P(more than 30 successes) = P(k > 30)
Use Φ to approximate binomial
probability in following examples
Suppose p=0.1, n=100
•  P(exactly 30 successes) = P(k = 30)
•  P(at least 30 successes) = P(k ≥ 30)
•  P(at most 30 successes) = P(k ≤ 30)
Comparing cumulative binomial with
normal distributions
Comparing binomial with normal
distributions (noncumulative)
Superimposition of binomial and
normal density graphs
Sampling distribution of the mean
•  Sampling distribution of the mean is the
distribution of sample means for all samples
of size n.
•  Sampling distribution of the mean is
approximately normal for large sample sizes
CENTRAL LIMIT THEOREM
• 
The mean of all sample means for samples of size n equals the
population mean.
–  for n=1, this is by definition of population mean
–  inductively, one can establish the truth for any fixed sample size
Die roll example: sampling WITH replacement, sample size of
2, compute average of sample variances
Die roll example: sampling WITHOUT replacement, sample
size of 2, compute average of sample variances
Note: 5.333 =(2+6+8)/3
Illustrates that mean of samples
equals population mean.
Sampling distribution of a statistic
•  More generally, the sampling distribution of
any statistic is the distribution of that statistic
over all samples of size n.
•  For instance, the sampling distribution of the
sample standard deviation (resp. variance) is
the distribution of all sample standard
deviations (resp. variances) over all samples
of size n.
Solution (first for sampling without
replacement)
Solution (second, for sampling
with replacement)
•  For more examples of this form, see my
demo python program
demoSampleMeansVar.py, in class web
pages. See also
•  outputDemoSampleMeansVarWithReplacement.tx
t and
•  outputDemoSampleMeansVarWithoutReplacemen
t.txt
Bear age histogram: Mean:43.52, Var: 1116.03
StDev:33.41 Max:177 Min:8
Histogram of the means of (X+X+X)/3 for ALL size 3 samples
of bear ages: Mean:43.52 Var: 372.01 StDev:19.29
Max:177 Min:8. NOTE that bear age stdev (previous
slide) divided by sqrt(3), where 3 is sample size, equals 19.29
Histogram of sample variances of ALL size 3 samples of
bear ages: Mean:1116.03
StDev:1567.25
Max:9520.33 Min:0. Note that the mean of all sample
variances equals the population variance.
Histogram of 10,000 repetitions of following experiment:
count number of 2’s when rolling die 100 times
Sampling distribution of the proportion
•  Sampling distribution of the proportion is the
distribution of all sample proportions for all
samples of (equal and fixed) size n
•  Proportion is number of “successes” over
number of “trials” (number of people who
plan to vote for candidate X over number of
voters, number of red balls over number of
balls in urn).
Properties of distribution of
sample proportions
•  sample proportions tend to target the value of
the propulation proportion
–  relative frequency of r red balls in sample of size n
is P(r/n) = h(n,r;N,R)
•  for sufficiently large sample sizes, the
distribution of sample proportions
approximates a normal distribution (recall that
density curve of hypergeometric is
approximately normal for sufficiently large
sample size.
Unbiased statistic
•  Unbiased statistic is a statistic that targets the
population parameter
•  sample mean, sample variance, sample
proportion are unbiased statistics that target
population mean, variance and proportion
•  median, range, stdev are statistics that do
not target corresponding population
parameters.
Central Limit Theorem: asymptotically, Z-scores
converge to standard normal distribution
Consequences of central limit theorem
•  Sampling distribution of the mean (which by
definition is the distribution of sample means
for all samples of equal and fixed size n) is
approximately normal for large sample sizes.
The expected value of the sampling
distribution for the mean is the population
mean.
•  Sampling distribution of the proportion is
approximately normal for large sample sizes.
The expected value of the sampling
distribution of the proportion is the population
mean.
However ...
•  Sampling distribution of the variance is NOT
necessarily approximately normal. In fact, we
saw that such an example with bear age
samples of size 3!
•  Nevertheless, the average of the sample
variances does equal the population variance,
provided that the sampling is done with
replacement.
Mean and stdev of sample means
Mean, variance of sample
distribution
•  Sampling with replacement (binomial):
–  Let X be r.v. counting number of successes in sample of size
n, where probability of success is p
–  E[X] = np, V[X] = npq
–  Let Y be r.v. counting relative frequency of success in
sample of size n
–  E[Y] = E[X/n] = np/n = p
–  V[Y] = V[X/n] =npq/n2 = pq/n = sigma2/n
–  stdev(Y) = sqrt(pq/n) = sigma/sqrt(n)
Mean, variance of sample
distribution
•  Sampling without replacement (hypergeometric):
–  Let X be r.v. counting number of successes in sample of size
n, where probability of success is p
–  E[X] = np, V[X] = npq(N-n)/(N-1)
–  Let Y be r.v. counting relative frequency of success in sample
of size n
–  E[Y] = E[X/n] = np/n = p
–  V[Y] = V[X/n] =V[X]2/n = (npq/n2)(N-n)/(N-1)
= (pq/n)(N-n)/(N-1)
–  stdev(Y) = stdev(X)/sqrt(n) = sqrt[ (pq/n)(N-n)/(N-1)]
Proportion of successes in n trials
•  Let X be r.v. that counts the number of successes in
n trials, where probability of success is p (absolute
frequency)
•  Let Y be r.v. that returns the proportion of successes
in n trials (relative frequency)
•  E[Y] = E[X/n] = E[X]/n = np/n = p
•  V[Y] = V[X/n] = V[X]/n2 = npq/n2 = pq/n
NOTE: We compute Z-score (x-mu)/(sigma/sqrt(n))
since we are working with sample distr of mean.
Now compute Φ(Z-score)
Normality test: quantile plot
•  sort data in increasing order x1,...,xn
•  for each i compute z-scores
NORMINV(i/n,0,1), or rather following
book NORMINV((2i-1)/2n,0,1)
•  determine if scatter plot (xi,zi) is linear
Quantile plot for
temperatures
Z-scores
3
2
y = 0.0304x - 1.6244
1
Zscores
0
1
-1
-2
-3
10
19
28
37
46
55
64
73
82
91 100
Linear (Zscores)
Interarrival time, or distance
between successive genomic
motifs
What probability distribution is this?
Exponential distribution continuous analogue of geometric
distribution
Image of exponential distribution for
different values of ¸ (called ® in previous
slide)
Excel statistical functions
• 
• 
• 
• 
• 
• 
• 
average
mode
stdev
stdevp
var
varp
max,min,median
Excel statistical functions
•  QUARTILE
–  Returns the quartile of a data set (quartile =
0,1,2,3,4)
–  Syntax: QUARTILE(array,quart)
•  PERCENTILE
–  Returns the k-th percentile of values in a range. For
example, you can decide to examine candidates
who score above the 90th percentile.
–  Syntax: PERCENTILE(array,k)
Here k is in interval [0,1] e.g. 0.9.
•  STANDARDIZE
–  Returns Z-score corresponding to x for a distribution with
mean and stdev.
–  Syntax: STANDARDIZE(x,mean,stdev)
•  BINOMDIST
–  Returns the individual term binomial
distribution probability.
–  Syntax: BINOMDIST(k,n,p,cumulative).
Used for sampling with replacement
Density function when cumulative = false
Cumulative density function when cumulative = true
•  HYPGEOMDIST
–  Returns the probability of a given number of
sample successes, given the sample size,
population successes, and population size.
–  Syntax: HYPGEOMDIST(r,n,R,N)
Used for sampling without replacement
•  PERMUT
–  Returns the number of ORDERED
sequences of length k drawn from a set of
size n. Permutations are different from
combinations, for which order is not
significant.
–  Syntax: PERMUT(n,k)
•  fact(k)
–  Returns the factorial k(k-1)(k-2)...1
–  use permut and fact to compute
combinations
•  POISSON
–  Returns the Poisson distribution. A common
application of the Poisson distribution is predicting
the number of events over a specific time, such as
the number of cars arriving at a toll plaza in 1
minute.
–  Syntax: POISSON(x,mean,cumulative)
•  NORMDIST
–  Returns the normal distribution for the specified mean and
standard deviation.
–  Syntax: NORMDIST(x,mean,stdev,cumulative)
•  NORMDIST
–  Returns the normal distribution for the specified mean and standard
deviation.
–  Syntax: NORMDIST(x,mean,stdev,cumulative)
•  NORMSINV
–  Returns the inverse of the standard normal cumulative distribution.
The distribution has a mean of zero and a standard deviation of one.
–  Syntax: NORMSINV(probability)
Tables from Book