Download computed from a sample of size n

8.1
Sampling Distributions: Distributions of the Sample Mean
HW 8.1: Add #20abc, #21abc from text
Review: samples, probability distribution, statistic, 𝑥̅
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Sampling distribution – a probability distribution for all possible values of a
statistic computed from a sample of size n.
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Sampling distribution of the sample mean – the probability distribution of all
possible values of the random variable 𝑥̅ computed from a sample of size n from a
population with mean μ and standard deviation σ.
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1st: Distribution of the sample mean (𝑥̅ ) – samples from a normal population.
Suppose that a simple random sample of size n is drawn from a large population
with mean μ and standard deviation σ. The sampling distribution of 𝑥̅ will have
𝜎
mean 𝜇𝑥̅ = 𝜇 and standard deviation 𝜎𝑥̅ = . (𝜎𝑥̅ is called the ‘standard error of
√𝑛
the mean).
Shape: If a random variable X is normally distributed, the distribution of the
sample mean, 𝑥̅ , is normally distributed.
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What if a population is not normal?
2nd: Distribution of the sample mean (𝑥̅ ) – samples from a population that is not
normal.
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Example: Dice: n=1, 2, 3
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The Central Limit Theorem
Regardless of the shape of the underlying population, the sampling distribution of
𝑥̅ becomes approximately normal as the sample size, n, increases.
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Rule of thumb: The distribution of the sample mean is approximately normal if the
sample size, n ≥ 30.
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The sampling distribution of 𝑥̅ will have mean 𝜇𝑥̅ = 𝜇 and standard deviation 𝜎𝑥̅ =
𝜎
.
√𝑛
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EX:
A simple random sample of size n = 36 is obtained from a population with μ = 64
and σ = 18.
(a) Describe the sampling distribution of 𝑥̅ .
(b) What is P(𝑥̅ > 62.6)?
Compare to P(x > 62.6).
(c) What is P(𝑥̅ < 68.7)?
(d) What is P(59.8 < 𝑥̅ < 65.9)?
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EX: p.439, #19
The length of human pregnancies is approximately normally distributed with mean
μ = 266 days and standard deviation σ = 16 days.
(a) What is the probability a randomly selected pregnancy lasts less than 260
days?
(b) Suppose a random sample of 20 pregnancies is obtained. Describe the
sampling distribution of the sample mean length of human pregnancies.
(c) What is the probability that a random sample of 20 pregnancies has a mean
gestation period of 260 days or less?
(d) What is the probability that a random sample of 50 pregnancies has a mean
gestation period of 260 days or less?
(e) What might you conclude if a random sample of 50 pregnancies resulted in a
mean gestation period of 260 days or less?