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```7.3 Notes pg. 451-461
(Feel free to add notes of your own.)
AP Exam Tip: Notation matters. The symbols
̂, 𝒙
̅, 𝝁, 𝝈, 𝒑, 𝝁𝒑̂, 𝝈𝒑̂, 𝝁𝒙̅, 𝝈𝒙̅ all have specific and different
𝒑
meanings. Either use notation correctly or don’t use it at all.
You can expect to lose credit if you use incorrect notation.
MEAN AND STANDARD DEVIATION OF A SAMPLE MEAN
• Suppose 𝑥̅ is the mean of an SRS size 𝑛 from a large population with µ and 𝜎.
• Mean of the sampling distribution of 𝑥̅ is 𝜇𝑥̅ = 𝜇
𝜎
• Standard deviation is 𝜎𝑥̅ =
𝜎
√𝑛
• Only use 𝜎𝑥̅ =
when the population is at least 10 times as large as the
√𝑛
sample or 𝑁 ≥ 10𝑛
• Sample mean 𝑥̅ is an unbiased estimator of the population mean 𝜇
• Values of 𝑥̅ are less spread out for larger samples
Example This Wine Stinks pg. 452
Skip the Think About It pg. 453
Sampling Distribution of a Sample Mean from a Normal Population
• Draw an SRS of size 𝑛 from a population that has the distribution 𝑁(𝜇, 𝜎).
• Then the sampling distribution of the mean 𝑥̅ has the distribution 𝑁 (𝜇,
𝜎
√𝑛
).
Read Example Young Women’s Heights pg. 455.
Central Limit Theorem
• Draw an SRS of size 𝑛 from any population whatsoever with mean 𝜇 and finite standard
deviation 𝜎.
• When 𝑛 is large, the sampling distribution of the sample mean 𝑥̅ is close to the
𝜎
normal distribution 𝑁 ( 𝜇, ) .
√𝑛
• Larger sample sizes are required if shape of the population is far from Normal, in
particular 𝒏 ≥ 𝟑𝟎.
Read Example Servicing Air Conditioners pg. 459
Collected HW: pg. 461-463 #54, 58, 66, 68 Due Wed. end of class Suggested problems: #49,57, 59, 65,67
```