Page 495 - Probability and Statistical Inference

472 9. Confidence Interval Estimation

average (µ) gas mileage per gallon. Assume a normal distribution for the gas
mileage.
9.2.15 The waiting time (in minutes) at a bus stop is believed to have an
exponential distribution with mean θ(> 0). The waiting times on ten occasions
were recorded as follows:
6.2 5.8 4.5 6.1 4.6 4.8 5.3 5.0 3.8 4.0

(i) Construct a 95% two–sided confidence interval for the true average
waiting time;
(ii) Construct a 95% two–sided confidence interval for the true vari-
ance of the waiting time;
(iii) At 5% level, is there sufficient evidence to justify the claim that the
average waiting time exceeds 5 minutes?
9.2.16 Consider a normal population with unknown mean µ and σ = 25.
How large a sample size n is needed to estimate µ within 5 units with 99%
confidence?
9.2.17 (Exercise 9.2.10 Continued) In the laboratory, an experiment was
conducted to look into the average number of days a variety of weed takes to
germinate. Twelve seeds of this variety of weed were planted on a dish. From
the moment the seeds were planted, the time (days) to germination was re-
corded for each seed. The observed data follows:
4.39 6.04 6.43 6.98 2.61 5.87
2.73 7.74 5.31 3.27 4.36 4.61

The team’s expert in areas of soil and weed sciences believed that the time to
germination had a Weibull distribution with its pdf

where a(> 0) is unknown but b = 3. Find a 90% two–sided confidence inter-
val for the parameter a depending on the minimal sufficient statistic.
9.2.18 (Exercise 9.2.17 Continued) Before the lab experiment was con-
ducted, the team’s expert in areas of soil and weed sciences believed that the
average time to germination was 3.8 days. Use the confidence interval found
in the Exercise 9.2.17 to test the expert’s belief. What are the respective null
and alternative hypotheses? What is the level of the test?
9.3.1 (Example 9.3.1 Continued) Suppose that the random variables
X , ..., X are iid i = 1, 2, and that the X ’s are independent of
1j
i
ini
the X ’s. Here we assume that the means µ , µ are unknown but σ , σ 2
1
2
1
2j
are known, (µ , σ ) ∈ ℜ × ℜ , i = 1, 2. With fixed α ∈ (0, 1), construct a
+
i
i