Exercises L4 Basic Stats –Applied statistics
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7.
X1, ..., X10 is a random sample of from a population having an expectation µ and a standard deviation σ.
Consider the following estimators of µ:
T1 = X1,
T2 = (X1 + X10 )/ 2,
T3 = X1+...+ X10 and T4 = ( X1+...+ X10 ) /10.
Use the Mean Squared Error,
, to determine which of these 4 estimators
of µ is the best.
Consider two independent samples from two different populations (having means µ 1 and µ2):
X1, ..., Xm is a random sample, drawn from a population having an expectation µ1 and standard deviation σ.
Y1, ..., Yn is a random sample, drawn from a population having an expectation µ2 and the same standard
deviation σ.
a. Give unbiased estimators for µ1 and µ2.
b. Give an estimator for the difference between these two means (µ1 - µ2 ) and show that it is unbiased
c. Compute the variance of the estimator in b.
A researcher wants to find out what the proportion of Gambian voters is who prefer Barack Obama as
president of the Gambia. He selects n = 800 voters to determine this proportion: X is the number of voters
amongst these 800 who prefer Obama.
a. Give the usual unbiased estimator for the population proportion and state its (approximating)
distribution.
b. Show that 0 ≤ p(1-p) ≤ ¼ (consider the functions f(x) = p(1-p) = p – p2 and its maximum value)
c. What is the maximum value of var( ?
A coffee dispenser is designed to supply a full cup of coffee (about 125 ml), but in practice the quantity
varies. The variance of the quantity is known: 5 ml2. An organisation for consumers tries to find the mean µ
of the quantity of a coffee poured by a type of coffee dispenser and measured the total quantity of 20 cups
of coffee: 2.42 liter.
a. What is the mean quantity (in ml) of coffee per cup?
b. Find a 95%-confidence interval for the mean µ. First state the statistical assumptions.
c. If we define the width of the interval as
, how wide will a 90%-confidence interval be if we
observe the quantity of n = 10, 20 or 30 cups?
In freezers in shops the temperature should be around -20 degrees Celsius. Here are 6 observations of the
temperature (on different moments) in the freezer of the supermarket Correct Choice:
-19.3, -20.1, -19.8, - 19.7, -20.6, -20.5
a. Compute the sample mean and the sample standard deviation of these temperatures (let your calculator
determine these values!).
b. Give the statistical assumptions which are necessary to compute confidence intervals for µ and σ.
c. Determine the 90%-confidence interval for the expected temperature.
d. Determine 90%-confidence intervals for both the variance and the standard deviation of the
temperatures in the freezer.
Consider the situation of exercise 3.
a. If the researcher found X = 328 voters in his sample who favour Obama as president of Gambia,
determine a 95%-Confidence Interval for this proportion in the population of all Gambians.
b. Repeat a. for confidence level 90%.
c. If we want the width of the interval in a. of this exercise to be 2% at most (e.g. (0.33, 0.35)), should the
researcher choose n greater or less than 800?
d. Use the result of exercise 3.b and c. to find the minimum value of n, such that the width of the 95%confidence interval is at most 2%.
IT-specialists in the Netherlands easily can find a job if they completed their education. A job consultant
wants to know what the monthly salary of starting IT-specialists is. In a survey he used a random sample of
15 of these specialists and found a mean of € 2500 (gross per month) and a standard deviation of € 600.
a. Compute a confidence interval for the salary of starting IT-specialists. Use a confidence level of 95%
and first state the statistical assumptions.
b. Give a proper interpretation of the interval computed in a.
8.
9.
c. Compute a 95%-confidence interval of the standard deviation of the salary of starting IT-specialists.
X1, ..., X9 is a random sample, drawn from a population having a N(µ, 9)-distribution.
We want to test H0 : μ = 10 and H1 : μ > 10 and take α = 0.05.
a. Find the rejection region if is the test statistic.
b. Compute the power of this test (=1 – P(type II error) ) for μ =11, μ = 12 and μ = 13 and sketch the
graph
c. If we observe = 12.5 find the p-value and draw a conclusion (verify that you will find the same result
using the rejection region at a.)
A consumers organisation is researching the life expectancy of Michelin tyres for four wheel drives. The
result of a random sample of 20 tyres is as follows: the sample mean is 41000 km and the sample variance is
1500 2 km2.Usually the tyres of trademarks have a life expectancy of 40000 km or less. Can Michelin claim
that the sample proofs their tyres live longer?
Use a significance level of 5% and report all 8 steps of the testing procedure.
10. The radioactive radiation in Europe was always around 130 m REM. After the major accident at the nuclear
power plant of Chernobyl scientists thought the radiation would increase. On 15 (random chosen) spots in
Belgium they observed the radiation (in m REM.)
313 112 218 166 288 151 249 348 87 203 233 133 182 267 52.
a. Compute mean and variation of these values
b. Find out, using a test, whether the radiation has increased in Belgium. Use the test procedure, α = 0.05
and decide using the rejection region.
c. Find (estimate) the p-value and compare the conclusion you can draw from it with the result of b.
11. Repeat exercise 8. a. and c. for the hypotheses H0 : μ = 10 and H1 : μ  10
12. A dietician wants to check the effect of a newly designed diet. A sample of 15 persons (having overweight)
Person
Before (kg)
After (kg)
13.
14.
15.
16.
1
98
93
2
81
82
3
107
102
4
85
87
5
82
88
6
150
94
7
110
112
8
90
95
9
81
75
10
100
91
11
97
94
12
75
78
13
101
93
14
94
85
15
98
90
is weighed before and after using the diet for two months. Here are the results:
a. First compute the loss of weight (Before minus after) for these 15 persons, and then the mean loss and
the sample variance of the loss of weight.
b. Does this enquiry show that the diet has a positive effect? (Report the 8 steps and use α = 0.05)
c. Find a 90%-confidence interval for the mean loss in weight.
A group of popular politicians started a new party “National Pride”, that will participate in coming
elections. The new party leader stated that he expected at least 50% of all votes: if the outcome of the
elections would be less than 50% he will resign. The day before election day a television program on
politics conducted a poll as to see whether the new party leader will resign. The poll concerned 1250 voters
and 575 (less than 50%) of them said they will vote “National Pride”.
a. Does this proof (beyond reasonable doubt) that the new party leader will have to resign when the
election results are known? Conduct a test and use a significance level of 1%.
b. Compute a 95%-confidence interval for the percentage of voters that will vote “National Pride”
A standard medicine cures 70% of all patients within a week. A medicine manufacturer developed a new,
cheaper medicine. In the testing phase it is used to cure a random sample of 200 patients: 148 were cured
within a week. Test the null hypothesis that the new medicine has the same success rate as the standard
medicine (the alternative hypothesis is that there is a difference). Take α = 0.10
Test for the data in exercise 4 whether the standard deviation of the temperatures is at most 1 degree Celsius
(α = 0.05)
For the situation in exercise 10 is furthermore given that before Chernobyl the variance of the radiation was
2500 (mREM2). Test at level 10% whether the variance of the radiation is changed