1. Ch. 8 Estimating with Confidence
Ch. 8-3 Estimating a Population Mean

2. a mean
Sample:

3. We want to estimate the true mean final exam score, 𝜇, for a class of AP Stats student at a 96% confidence level.
one sample z interval for mean
Random:
The sample is an SRS.
Normal:
The population distribution is normal, so the sampling distribution of 𝑥 is approximately normal.
Independent:
Sampling without replacement so check 10% condition
There are at least 10(4) = 40 AP Stats students.

4. Estimate ± Margin of Error
𝜎 𝑛
𝑥 ± 𝑧 ∗
.96
.02
.02
_____ ±
invNorm(.98, 0, 1) = 2.054
𝑧 ∗ =2.054
_____ ± 4.457
With calculator:
_________, _________
STAT  TESTS  ZInterval (7)
Inpt: 𝜎:
𝑥 : n:
C-Level:
Data Stats
4.34
_____
_________, _________ 4
0.96
We are 96% confident that the interval from ______ to ______ captures the true mean final exam score, 𝜇, for the class of AP Stats students.

5. = 𝑧 ∗ 𝜎 𝑛
96% confidence level → 𝑧 ∗ = 2.054
Margin of Error
4= 𝑛
1.95= 𝑛
1.95 𝑛 =4.34
𝑛 =2.226
𝑛=4.96
We still round up.
𝑛=5

6. 𝜎
To know 𝜎, we would have to know all of the test
scores. If we knew them, we would simplify find 𝜇 rather than estimate it.
𝑠 𝑥
𝑡-score
𝑡-distribution
𝑥 −𝜇 𝜎 𝑛
𝑥 −𝜇 𝑠 𝑥 𝑛
From now on, use 𝑧 for proportions and 𝑡 for means (if 𝜎 is unknown)
𝑧 𝑎 𝑝 𝑡 𝑎 𝑥

7. A 𝑡-distribution has a different shape than the Standard Normal Curve
A 𝑡-distribution has a different shape than the Standard Normal Curve. It’s still symmetric with a peak at 0 but has much more area in the tails.
The shape of a 𝑡-distribution depends on the sample size, which indicates its degrees of freedom:
df = 𝑛−1→ 𝑡 𝑛−1
𝑡 has the same interpretation as 𝑧, “how many std dev 𝑥 is from 𝜇.”

8. 𝒅𝒇=𝒏−𝟏
more
more
more
closer
more

9. SAME
one-sample T interval for mean
conditions SAME

10. wider than the 𝑧-interval calculated before.
Estimate ± Margin of Error
𝑡-distribution
𝑠 𝑥 𝑛
𝑥 ± 𝑡 ∗
.96
.02
.02 4
_____ ±
Can’t use invNorm since we’re using a 𝑡-distribution.
We now use Table B/𝑡-table (next slide)
_____ ± _______
_________, _________
𝑑𝑓=𝑛−1=4−1=3
wider than the 𝑧-interval calculated before.
So 𝑡 ∗ =3.482
Or use invT(area on left, df)
invT(0.98, 3) = 3.482
With calculator:
STAT  TESTS  TInterval (8)
Inpt:
𝑥 :
𝑆 𝑥 : n:
C-Level:
Data Stats
_____
_____
_________, _________ 4
0.96

11. So 𝑡 ∗ =3.482 Notice table gives area on the right
Have to look at 3 things:
1) Upper tail prob (.02)
2) 𝑑𝑓=4−1=3
3) Conf level (96%)
So 𝑡 ∗ =3.482

12. We are 96% confident that the interval from ______ to ______ captures the true mean final exam score, 𝜇, for the class of AP Stats students.

13. Practice Quiz for 10 min!
Trade papers and grade.
1.497
We want to estimate the true mean lap time, 𝜇, for Mr. Brinkhus’ career at MB2 Raceway at a 95% confidence level.
one sample T interval for mean
Random:
“Think of 9 laps as a SRS.”
Normal:
The population distribution is normal, so the sampling distribution of 𝑥 is approximately normal.
Independent:
Mr. Brinkhus did more than 10 9 =90 laps.

14. Estimate ± Margin of Error
𝑡-distribution
𝑠 𝑥 𝑛
𝑥 ± 𝑡 ∗
.95
.025
.025 ±
𝑑𝑓=𝑛−1=9−1=8
±1.151
𝑡 ∗ =invT 0.975, 8 =2.306
99.493,
We are 95% confident that the interval from to captures the true mean lap time, 𝜇, for Mr. Brinkhus’ career at MB2 Raceway.

15. 112.12
86.14, 138.1
With calculator:
36.32
STAT  TESTS  TInterval (8)
Inpt:
𝑥 :
𝑆 𝑥 : n:
C-Level:
Data Stats
112.12
36.32 10
0.95
Notice the interval got a lot wider because of the outlier.

16. Sample more laps! Make 𝑛≥30.
Normal – pop dist may not be normal so sampling dist may not be normal
cannot
ROBUST
accurate
Sample more laps! Make 𝑛≥30.

17. 𝑛≥30
CLT
Central Limit Theorem
outliers
strong skewness
You MUST graph the sample data (using a dotplot or boxplot) to see if the data is approximately normal.

18. Since the sample of 50 students was only from your classes and not from all students at your school, we should not use a 𝑡 interval to generalize about the mean GPA for all students at the school. Not an SRS.
Since the distribution is only moderately skewed and the sample size is larger than 15, it is safe to use a 𝑡 interval.
Since the sample size is small and there is a possible outlier, we should not use a 𝑡 interval.