Chapter 5: estimation techniques and sample analysis môn Xác suất thống kê| Trường Đại học Ngoại Thương

CHAPTER 5. ESTIMATION
1. Two samples of size n = 3 and n = 4 are drawn from a population distributed A(p).
a. Show that the sample frequencies f1 and f2 are both unbiased estimators of p
b. Which estimate is more efficient? Find the efficiency of f2 relative to f1.
c. In the class of linear estimators a.f1 + (1– a).f2 find a more efficient estimator of p.
2. Suppose two economists estimate the average household expenditure on food and they
use two unbiased (and independent) estimates, U and V. Because the second economist is
less careful. Therefore, the standard deviation of V is three times larger than the standard
deviation of U. To combine the two estimates to obtain a common estimate of m, three ways are suggest2ed 𝑈 + : 1 1= 1 2𝑉 (i) 𝑊 4𝑈 + 1 2= 3 4𝑉 (ii) 𝑊 (iii) 3= 1. 𝑈 + 0. 𝑉 𝑊
a. Which of the above estimates is unbiased? b. Which of the above estimates is efficient (most efficient).
Estimation the population mean
3. A store's sales are a normally distributed random variable with a standard deviation of 2
million/month. Randomly surveying the sales of 600 similarly sized stores found an average
sales of 8.5 million. With 95% confidence, estimate the average sales of stores of that size.
4. With 95% confidence, estimate the average fuel consumption for a car traveling from A
to B, if 30 test runs on this road segment are recorded, the amount of fuel consumed is as follows: the fuel consumption (litre) Number 9,6 – 9,8 3 9,8 – 10,0 5 10,0 – 10,2 10 10,2 – 10,4 8 10,4 – 10,6 4 n = 30
5. Know that the amount of fuel consumed is a random variable that obeys the normally
distribution. To determine the average price for a commodity in the market, people randomly
survey at 100 stores to obtain the following table of data: Giá (đồng)
83 85 87 89 91 93 95 97 99 101 Số cửa hàng 6 7 12 15 30 10 8 6 4 2
6. The length (X) of product type A produced by an automaton is a random variable that
follows the normal distribution with standard deviation of 3cm. To estimate with confidence
0.99, how many products must be measured when the symmetric confidence interval length does not exceed 0.6cm.
7. The weight (X) of a part is a random variable normally distributed with 1.2 kg. At least
how many details must be selected to investigate, if we want the accuracy of the estimate to
be 0.3 and the confidence of the estimate to be 0.95.
8. The yield of maize variety A in an area is reported through 25 harvest points and the results are as follows: The yield (x100kg/ha) Number 7 2 9 7 11 12 13 3 15 1 17 n = 25
With 95% confidence, calculate the minimum average yield for this region, knowing that
the maize yield for this region is a normally distributed random variable.
9. To determine the processing time of a machine part, people randomly monitor the
processing process of 25 parts and obtain the following table of data: Time (minutes) Number 14 2 16 6 18 11 20 4 22 2 24 n = 25
With a confidence level of 0.95, estimate the maximum average machining time for that
part. Assume that the part machining time is a normally distributed random variable. Estimated population rate
10. Trial opening 200 boxes of a cannery; It is found that there are 8 metamorphic boxes.
With a confidence level of 0.95, estimate the percentage of canned goods that are depot metamorphosed.
11. Sowing 400 seeds, 20 seeds did not germinate. What is the maximum percentage of
seeds that do not germinate? Requires a conclusion with a confidence level of 95%.
12. During the presidential election campaign, 1600 voters were randomly interviewed and
found that 960 of them will vote for candidate A. With 99% confidence, candidate A will
win at least what percentage of votes?
13. The germination rate of a seed is 90%. It is necessary to estimate the germination rate
of that seed with a confidence level of 0.95 and a confidence interval length of not more
than 0.02, how many seeds should be sown?
14. In 1990, when randomly interviewing 1,500 people, the smoking rate was 43%. After a
long period of campaigning not to smoke, interviewing 1,000 people in 1997 found that the
smoking rate was 38%. With 95% confidence, estimate the percentage of smokers that has
decreased after 7 years of exercise.
Estimate the population variance
15. The stock return of each company over the past 5 years has been 15%, 10% 20% 7%
14%. With 90% confidence, estimate the dispersion of that company's stock return. We
know stock returns are a normally distributed random variable
16. To study the fluctuation of milk volume of cows during the milking cycle, 15 cows were
randomly selected and obtained the following data (unit: liters) 12298 13812 11036 12120 14358 9248 14972 8989 9980 14004 10620 11990 14788 14744 14786
With 95% confidence, estimate the variation in milk production per cow during the
milking cycle. Knowing the milk volume of cows is random normally distributed.
17. To study the stability of a machining machine, people randomly take 25 parts processed
by that machine, measure and obtain the following dimensions: 24,1 27,2 26,7 23,6 26,4 25,8 27,3 23,2 26,9 27,1 22,7 26,9 24,8 24,0 23,4 24,5 26,1 25,9 25,4 22,9 26,4 25,4 23,3 23,0 24,3
With 95% confidence, estimate the maximum dispersion of the dimensions of the parts
processed by that machine. Knowing the part size to be machined is a normally distributed random variable.