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AP Stats
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Ch 23 day 1: Confidence Intervals with Means
Example: A coffee machine dispenses coffee into paper cups. You’re supposed to get 10 ounces of coffee,
but the amount varies slightly from cup to cup. Here are the amounts measured in a random sample of 20
cups. Is there evidence that the machine is shortchanging customers?
9.9
9.7
10.0
9.9
9.6
9.8
10.0
9.5
9.7
9.9
9.6
10.2
10.0
9.9
9.5
How is this situation different from our previous examples?
10.1
9.8
10.1
9.8
9.9
How do proportions compare with means?
Proportions
Means
Measure of Center:
 Population’s Parameter
 Sample’s Statistic
Measure of Spread:
 Population’s Parameter
 Sample’s Statistic
Distribution Model for Sample’s
Measure of Center
(
Confidence Interval form
William Gosset was in charge of quality control for Guinness Brewery in Dublin in the early 1900’s.
He found that current statistical methods were causing them to throw out perfectly good beer. He
took a leave of absence to simulate this problem and came up with a more accurate way to make
decisions about the mean of a population from a few samples. The Guinness Company would not let
him associate his name with his work, so it was named: Student’s t distribution. How does the
Student’s t distribution compare to the Normal z distribution?
Impact of sample size n:
Standard Deviation:
Confidence Interval’s range:
Back to our coffee example. Let’s calculate a 95% confidence interval for the average amount of coffee
dispensed and see if our belief (getting short changed) holds water:
Conditions to check:
1. Plausibly independent? (why)
2. Randomly selected? (why)
3. 10% condition? (why)
4. **Nearly Normal Condition** (remember me?)
Mechanics:
Conclusion:
A medical researcher measured the body temperature of a randomly selected group of adults. The data he
collected are summarized below from some computer output. We wish to estimate the average
(“normal”) temperature among the adult population.
Statistic
Temperature
Count
52
Mean
98.285
Median
98.200
Std Dev
0.6824
Task:
Parameter:
Population:
Conditions to use Students t distribution to model this sample’s mean:
1. Plausibly independent?
2. Randomly selected?
3. Independent or 10% Condition?
4. Data in sample: Symmetrical, unimodal, w/ no outliers? (Histogram if possible)
Mechanics:
Conclusion:
IQR
1.050
Homework: Ch.23 pp.541-546 #12, #28, #34 - use extra paper and the space you
need to answer each question.
12. Parking. Hoping to lure more shoppers downtown, a city builds a new public parking garage in the
central business district. The city plans to pay for the structure through parking fees. During a twomonth period (44 weekdays), daily fees collected averaged $126, with a standard deviation of $15.
a)
What assumptions must you make in order to use these statistics for inference?
b)
Write a 90% confidence interval for the mean daily income this parking garage will generate.
c)
Explain in context what this confidence interval means.
d)
Explain what “90% confidence” means in this context.
e)
The consultant who advised the city on this project predicted that parking revenues would average
$130 per day. Based on your confidence interval, do you think the consultant could have been
correct? Why?
28. Doritos. Some students checked 6 bags of Doritos marked with a net weight of 28.3 grams. They
carefully weighed the contents of each bag, recording the following weights (in grams):
29.2, 28.5, 28.7, 28.9, 29.1, 29.5.
a)
Do these data satisfy the assumptions for inference? Explain.
b)
Find the mean and standard deviation of the observed weights.
c)
Create a 95% confidence interval for the mean weight of such bags of chips.
d)
Explain in context what your interval means.
e)
Comment on the company’s stated net weight of 28.3 grams.
34. Yogurt. Consumer Reports tested 14 brands of vanilla yogurt and found the following numbers of
calories per serving:
160
200
220
230
120
180
140
130
170
190
80
120
100
170
a)
Check the assumptions and conditions for inference.
b)
Create a 95% confidence interval for the average calorie content of vanilla yogurt.
c)
A diet guide claims that you will get 120 calories from a serving of vanilla yogurt. What does this
evidence indicate? Use your confidence interval to test an appropriate hypothesis and state your
conclusion.