Chapter 23
Inference for OneSample Means
Steps for doing a confidence interval:
1)
2)
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State the parameter
Conditions
1) The sample should be chosen randomly
2) The sample distribution should be approximately normal
- the population is known to be normal, or
- the sample size is large (n  30), or
- graph data to show approximately normal
3) 10% rule – The sample should be less than 10% of the
population
4) σ will almost always be unknown
3) Calculate the interval
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If σ is unknown we perform a t-interval…the t distribution
is based on an unknown standard deviation and different
sample sizes (known as degrees of freedom)
4) Write a statement about the interval in the context
of the problem.
Formula for a t-confidence interval:
Critical value
 s 
x  t *

 n
estimate
Margin of error
Standard
deviation of
statistic
df  n  1
Degrees of
freedom
In a randomized comparative experiment
on the effects of calcium on blood
pressure, researchers divided 54 healthy,
white males at random into two groups,
takes calcium or placebo. The paper
reports a mean seated systolic blood
pressure of 114.9 with standard deviation
of 9.3 for the placebo group. Find a 95%
confidence interval for the true mean
systolic blood pressure of the placebo
group.
State the parameters
μ = the true mean systolic blood pressure of healthy white males
Justify the confidence interval needed (state assumptions)
1) The sample must be random which is stated in the problem.
2) The sample distribution should be approximately normal.
Since n = 54 >30, by the CLT we can assume the sample
distribution is approximately normal.
3) The sample should be less than 10% of the population.
The population should be at least 540 healthy white males,
which I will assume.
4)  is unknown
Since the conditions are satisfied a t – interval for means
is appropriate.
Calculate the confidence interval.
x  114.9
 s
x t
n  54
n
s  9.3
 9.3 
95% CI
df  53

114.9  2.006
 54 
112.36,117.44
Explain the interval in the context of the problem.
We are 95% confident that the true mean systolic
blood pressure for healthy white males is between
112.36 and 117.44.
Steps for a hypothesis test :
1) Define the parameter
2) Hypothesis statements
3) Assumptions
4) Calculations (Find the p-value)
5) Decision and Conclusion in
context
Conditions for one-sample means
1) The sample should be chosen randomly
2) The sample distribution should be approximately normal
- the population is known to be normal, or
- the sample size is large (n  30), or
- graph data to show approximately normal
3) 10% rule – The sample should be less than 10% of the
population
4) σ will almost certainly be unknown, we perform t-test
Formulas:
 unknown:
statistic - parameter
test statistic 
standard deviation of statistic
t=
x
s
n
x
df  n  1
Example In 1998, as an advertising campaign, the
Nabisco Company announced a “1000 Chips Challenge,”
claiming that every 19-ounce bag of their Chips Ahoy
cookies contained at least 1000 chocolate chips.
Dedicated Statistics students at the Air Force Academy
(no kidding) purchased some randomly selected bags of
cookies, and counted the chocolate chips. Some of their
data are given below.
1219 1214 1087 1200 1419 1121 1325 1345
1244 1258 1356 1132 1191 1270 1295 1135
What does this say about Nabisco’s claim? Test an
appropriate hypothesis at 5% significance.
Parameters and Hypotheses
μ = the true mean number of chocolate chips in each bag of
Chips Ahoy
H0: μ = 1000
Ha: μ > 1000
Assumptions (Conditions)
1) The sample must be random which is stated in the problem.
2) The sample distribution should be approximately normal. (Check with an
appropriate graphical display)
The boxplot shows no outliers, so assume that the sample distribution
is approx. normal.
3) The sample should be less than 10% of the population. The population
should be at least 160 bags of Chips Ahoy, which we will assume.
4)  is unknown
Since the conditions are met, a t-test for the one-sample means is appropriate.
Calculations
x  x
 x    1000 t  s 
n
x  1238.1875
n  16
s  94.282
 = 0.05
df  15
 (1000)
1238.1875
94.282
 10.1053
16
p  value  P(t  10.1053)  2.176 108
8
.2.176 10  .05
Decision: Since p-value < , I reject the null hypothesis at the .05
level.
Conclusion:
There is sufficient evidence to suggest that the true mean number
of chocolate chips in each bag of Chips Ahoy is greater than 1000.