Sec. 7.3, cont.: Sample Means and Central Limit Theorem
What happens when x is not normal?
Central Limit Theorem: Draw an SRS of size n from any finite population whatsoever with mean µ and
standard deviation σ. When n is large, the sampling distribution of the sample mean x is close to the
normal distribution N(µ,

n
) with mean µ and standard deviation

n
.
Rule of thumb: n ≥ 30
Normal / Large Sample Condition for Sample Means

If the population distribution is Normal, then so is the sampling distribution of 𝑥̅ . This is true no matter
what the sample size n is.

If the population distribution is not Normal, the central limit theorem tells us that the sampling
distribution of 𝑥̅ will be approximately Normal in most cases if n ≥ 30.
Example, p. 459
Your company has a contract to perform preventive maintenance on thousands of air-conditioning units in a
large city. Based on service records from the past year, the time (in hours) that a technician requires to
complete the work follows a strongly right-skewed distribution is µ = 1 hour and σ = 1 hour. In the coming
week, your company will service an SRS of 70 air-conditioning units in the city. You plan to budget an average
of 1.1 hours per unit for a technician to complete the work. Will this be enough?
What is the probability that the average maintenance time 𝑥̅ for 70 units exceeds 1.1 hours? Show your work.
The sampling distribution of the sample mean time 𝑥̅ spent working on 70 units has
 Mean: 𝜇𝑥̅ = 𝜇 = 1 hour

1
Check 10% condition: 70 ≤ 10· air-conditioning units in population; Safe to assume
there are more than 700 air-conditioning units in the population, so
Standard deviation: 𝜎𝑥̅ =
𝜎
√𝑛
=
1
√70
≈ 0.12 hour
 Approximately Normal by Central Limit Theorem: 70 ≥ 30.
So 𝑥̅ ~N(1, 0.12)
P(𝑥̅ > 1.1) = P(𝑧 >
1.1−1
0.12
) = P(z > 0.83) = 1 – 0.7967 = 0.2033
Conclusion: If you budget 1.1 hours per unit, there is about a 20% chance that the technicians will not
complete the work within the budgeted time. You will have to decide if this risk is worth taking or if
you should schedule more time for the work.
HW: p. 462 # 59, 63, 65,
Due: Friday