7.5 Notes 7.5 – The Central Limit Theorem
The x̅ distribution, given x is normal.
x vs x̅
Theorem 7.1 for a Normal Probability
Distribution
Let x be a random variable with a normal
distribution whose mean is &micro; and whose standard
deviation is ơ. Let x̅ be the sample mean
corresponding to random size n taken from the x
distribution. The following true are true:
1. The x̅ distribution is a normal distribution.
2. The mean of the x̅ distribution is &micro;
3. The standard deviation of the x̅ distribution is
ơ
𝑛
So…
We can convert the x̅ distribution to the
standard normal z distribution.
&micro;𝑥̅ = &micro;
ơ𝑥̅ =
z =
𝑥̅ − &micro;
ơ
=
ơ
𝑛
𝑥̅ − &micro;𝑥̅
ơ𝑥̅
=
𝑥̅ − &micro;
ơ/ 𝑛
Vocabulary / Terminology
𝑺𝒕𝒂𝒏𝒅𝒂𝒓𝒅 𝒆𝒓𝒓𝒐𝒓 𝒐𝒇 𝒕𝒉𝒆 𝒎𝒆𝒂𝒏 ∶ ơ𝑥̅ =
ơ
𝑛
Central Limit Theorem 7.2: x can have any
distribution, but as the sample size gets larger
and larger, the distribution of x will approach a
normal distribution.
• Large sample: n ≥ 30
Guided Exercise #10
• As whole group, turn to page 290
– Whole group clarification
Find Probabilities Regarding x̅
1. Must be normally distributed
– If x distribution is normal, then x̅ is normal
– If x distribution is not normal, but the sample size n ≥
30, then x̅ is approximately normal
2. Convert x̅ to z using formula
z =
𝑥̅ − &micro;𝑥̅
ơ𝑥̅
=
𝑥̅ − &micro;
ơ/ 𝑛
3. Use the standard normal distribution (table 3) to
find corresponding probabilities regarding x̅
Guided Exercise #11
• As whole group, turn to page 292