The Normal Distribution

1
The Normal Distribution
2
n 20,290 ? 2622.0 ? 2037.9
Population
3
Y 2767.2 s 2044.7
Y 2675.4 s 1539.2
SAMPLES
Y 2588.8 s 1620.5
Y 2702.4 s 1727.1
4
Sampling distribution of the mean
Y 2767.2 s 2044.7
Y 2675.4 s 1539.2
Y 2588.8 s 1620.5
Y 2702.4 s 1727.1
5
Sampling distribution of the mean
1000 samples
6
Sampling distribution of the mean
7
Non-normal
Sampling distribution of the mean
Approximately normal
8
(No Transcript)
9
Sample means are normally distributed

• The mean of the sample means is m.
• The standard deviation of the sample means is

If the variable itself is normally
distributed, or sample size (n) is large
10
Standard error

• The standard error of an estimate of a mean is
  the standard deviation of the distribution of
  sample means

We can approximate this by
11
Distribution of means of samples with n 10
12
Larger samples equal smaller standard errors
13
Central limit theorem
14
Button pushing times
Frequency
Time (ms)
15
Distribution of means
16
Binomial Distribution
17
Normal approximation to the binomial distribution
18
Example
A scientist wants to determine if a loonie is a
fair coin. He carries out an experiment where
he flips the coin 1,000,000 times, and counts the
number of heads. Heads come up 543,123 times.
Using these data, test the fairness of the loonie.
19
Inference about means
Because is normally distributed
20
But... We dont know s
A good approximation to the standard normal is
then
Because we estimated s, t is not exactly a
standard normal!
21
t has a Students t distribution

22
Degrees of freedom
Degrees of freedom for the students t
distribution for a sample mean
df n - 1
23
Confidence interval for a mean
24
Confidence interval for a mean
?(2) 2-tailed significance level Df degrees
of freedom SEY standard error of the mean
25
95 confidence interval for a mean
Example Paradise flying snakes
Undulation rates (in Hz)
0.9, 1.4, 1.2, 1.2, 1.3, 2.0, 1.4, 1.6
26
Estimate the mean and standard deviation
27
Find the standard error
28
Table A3.3
29
Find the critical value of t
30
Putting it all together...
31
99 confidence interval
32
Confidence interval for the variance
33
(No Transcript)
34
Table A3.1
35
95 confidence interval for the variance of
flying snake undulation rate
36
95 confidence interval for the standard
deviation of flying snake undulation rate
37
One-sample t-test
38
Hypotheses for one-sample t-tests
H0 The mean of the population is m0. HA The
mean of the population is not m0.
39
Test statistic for one-sample t-test
m0 is the mean value proposed by H0
40
Example Human body temperature
H0 Mean healthy human body temperature is
98.6ºF HA Mean healthy human body temperature
is not 98.6ºF
41
Human body temperature
42
Degrees of freedom
df n-1 23
43
Comparing t to its distribution to find the
P-value
44
A portion of the t table
45
(No Transcript)
46
-1.67 is closer to 0 than -2.07, so P gt a With
these data, we cannot reject the null hypothesis
that the mean human body temperature is 98.6.
47
Body temperature revisited n 130
48
Body temperature revisited n 130
t is further out in the tail than the critical
value, so we could reject the null hypothesis.
Human body temperature is not 98.6ºF.
49
One-sample t-test Assumptions

• The variable is normally distributed.
• The sample is a random sample.