In class examples:
Answer: 7.7 to 8.3 pH
c) Answer: .9312
The Central Limit Theorem
Central limit theorem: When randomly sampling from any population with mean m and
standard deviation s, when n is large enough, the sampling distribution of y̅ is
approximately normal: N(m,s/√n).
• The larger the sample size n, the better the approximation of normality.
• This is very useful in inference: Many statistical tests require normality for the sampling
distribution. The central limit theorem tells us that, if the sample size is large enough,
we can safely make this assumption even if the raw data appear non-normal.
How Large a Sample Size?
It depends on the population distribution. More observations are required if the
population distribution is far from normal.
• A sample size of 25 or more is generally enough to obtain a normal sampling distribution
from a skewed population, even with mild outliers in the sample.
• A sample size of 40 or more will typically be good enough to overcome an extremely
skewed population and mild outliers in the sample
How do we know if the Population is Normal?
• Sometimes we are told that a variable has an approximately normal distribution (e.g.
large studies on human height or bone density).
• Most of the time, we don’t know and we only have sample data.
o We can summarize the data with a histogram and describe its shape. If the
sample is random, the shape of the histogram should be similar to the shape of
the population distribution.
o If the histogram appears non-normal, the central limit theorem informs us
whether the sampling distribution should look roughly normal or not.
1 Stat 1400
Population with strongly Sampling distribution of 𝑦 ത
skewed distribution for n = 2
The larger the sample size (n)
the more normal the
distribution curve appears
Sampling distribution of 𝑦 ത Sampling distribution
for n = 10 of 𝑦 for n = 25
Examples:ram of big toe deformations
12 Angle of big toe deformations in 38 patients:
• Symmetrical, one small outlier
• Population likely close to normal
u 6 • Sampling distribution ~ normal
10 15 20 25 30 35 40 45 50 More
Histogram of # of fruit per day for 74 girls
• Skewed, no outlier
• Population likely skewed
• Sampling distribution ~ normal given
large sample size
2 Stat 1400
Atlantic acorn sizes (in cm ) 14
Sample of 28 acorns: 12
Describe the distribution of the sample. 10