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Home , Sampling distribution

UNIVERSITY OF VICTORIA Midterm June 2014 Solutions

NAME: ______

STUDENT NUMBER: V00______

Course Name & No. Inferential Economics 246 Section(s) A01 CRN: 31175 Instructor: Betty Johnson

Duration: 1hour 50 minutes

This exam has a total of _8_ pages including this cover page. Students must count the number of pages and report any discrepancy immediately to the Invigilator. This exam is to be answered: In Booklets provided

Marking Scheme: 1. 5 marks 2. 5 marks 3. 5 marks 4. 5 marks 5. 5 marks 6. 5 marks 7. 10 marks 8. 10 marks 9. 5 marks 10. 20 marks

Materials allowed: Non-programmable calculator

2 Question 1: (5 marks) If the population σ is equals 96 and the size n=15, the variance of X is:

2 V( X )=σ / n= 96/15= 6.4

Question 2: (5 marks) Suppose ‘W’ is the width of tennis rackets used by all players on the university team. Suppose W ~N(16, 9). A random sample of 16 players is drawn from this population. What is the probability that the sample average team racket width is more than 14 cm?

Z=[14-16]/[3/4]= -2/.75 = -2.67 P(Z>-2.67)= P(Z< 2.67) =0.9962

Question 3: (5 marks) Assume the is from a . Given that σ 2=25, n=5, use the Chi- squared distribution to determine the probability that the sample variance is greater than 56. Assume the data is from a normal distrsibution. Use the Chi-square table to solve.

2 2 ()51− s ()() 456 χ = ==896. 4 σ 2 25

2 2 Ps[][]≥=56 Pχ4 ≥ 8. 96 010...to 005

Using the Chi-square table there is no specific value for 8.96. But, it is between 10% and 5%.

Question 4: Why does the sample size play such an important role in reducing the of the ? What are the implications of increasing the sample size? (5 marks)

ANSWER: The standard error is the of the population you are from divided by the standard deviation of the sample size. So, mathematically as the sample size increases, the standard error naturally decreases. But there is more to this, because the standard error is the standard deviation of the population of sample . So, as the sample size increases, the sample means are deviating less and less from the true population mean. Hence, as we sample more, we get statistics which are closer to the true parameters and our inference methods will improve. This is true for sampling distributions of mean, proportions, and .

Question 5: Describe the . Illustrate your answer with an example or examples. Total marks:5 Regardless of the form of the population, as the sample size increases, the sampling distribution will be approximately normal. A normal population will generate a normal sampling distribution.

“Regardless of the distribution of the parent population, as long as it has a finite mean µ and variance σ2, the distribution of the means of the random samples will approach a normal distribution, with mean μ and variance σ2/n, as the sample size n, goes to infinity.”

(I) When the parent population is normal, the sampling distribution of X is exactly normal.

(II) When the parent population is not normal or unknown, the sampling distribution of X is approximately normal as the sample size increases.

Question 6: Describe the concept of . Illustrate the technique with an example. Total marks:5

“The use of stratified sampling requires that a population be divided into homogeneous groups called strata. Each stratum is then sampled according to certain specified criteria.” Under sampling with prior knowledge. Divide population into strata. Each strata is different. Elements in the strata are the same. Sample each strata to replicate the same socio-economic situation as the population. Sampling is random within each strata. Example: If we divide students into two strata: residents and non-residents, and then sample in the same proportion, we may get a better average student loan estimate.

65% of university students 35% are from other parts of the are residents of the city province

Question 7: Describe and illustrate the four properties of a good estimator.

(I) Unbiasedness:

On average, the value of the estimate should equal the population parameter being estimated.

If the average value of the estimator does not equal the actual parameter value, the estimator is a biased estimator.

Ideally, an estimator has a bias of zero if it is said to be unbiased: f (θ$) ~ f θ ~ (f (θ)$) f (θ )

~ ~ E(θθ$) = E(θθ) ≠ θθ$, ~ ~ Bias θθθ=−[(E )]

(II) : The most efficient estimator among a group of unbiased estimators is the one with the smallest variance (or dispersion of values).

(III) Sufficiency: AAn estimator is said to be sufficient if it uses all the information about the population parameter that the sample can provide.@

Estimator incorporates all of the information available from the sample.

Sufficient estimators take into account each sample observation and any information that is generated by these observations.

(IV) Consistency: >Large Sample Property=

Usually the distribution of an estimator will change as the sample size changes.

(The sample size changes the distribution:

The properties of estimators for large sample sizes (as n N or infinity) are important. (Biasness and inefficiency of estimators may change as n approaches infinity.)

Properties of estimators based on distributions approached as n becomes large, are called asymptotic properties.

These properties may differ from the finite or small sample properties.

Consistency is the most important asymptotic property: A achieves convergence in probability limit of the estimator to the population parameter as the size of n increases. (Beyond this course.)

What we will discuss is a >stronger= notion of consistency: Mean Square Consistency:

Recall: MSE= variance + bias2.

An estimator is mean square consistent if its MSE 0 as the sample size, n, becomes large.

AAn estimator, θ$ θ$ , is mean square consistent if its MSE:

E (θ$ -θ )2, approaches zero as the sample size becomes large@.

EandVasn()θθ$$→→→∞ () θ 0 .

Note: If an estimator is mean square consistent, then it will also be consistent in the convergence in probability sense; But an estimator may be consistent in the convergence in probability sense, yet not be mean square consistent.

Consistency implies that the probability distribution of the estimator for large samples becomes smaller and smaller (i.e. variance is decreasing as more information about the population is used in each sample). The distribution becomes more centred about the true value of the parameter (bias getting smaller). And in the limit as n = ∞ , the probability distribution of the estimator degenerates into a single Aspike@ at the true value.

Question 8: Total marks:10 (i) Using the fact that the mean of the chi-squared distribution is (n-1), prove that ES()22= σ

Es()22= σ Since En()χ 2 =−1 ()ns−1 2 and χ 2 = σ 2 if you take the expectation: ⎡()ns−1 2 ⎤ E⎢ 2 ⎥ =−n 1 ⎣ σ ⎦ n −1 Es22= σ [] n −1 Es[]22= σ

(ii) Prove that . EX()= μ 2 Let Xi ~( μ X, σ ) for all i.

Since : (i)

1 X =+++()XX X n 12L n and (ii) E (Xi) = µ ,

we can apply the rules of expectation:

⎛ 1 n ⎞ EX()= E⎜ ∑ X i ⎟ ⎝ n i=1 ⎠ 1 ⎛ n ⎞ = EX⎜ ∑ i ⎟ n ⎝ i=1 ⎠ 1 =+++EX()12 X Xn n L 1 =++++EX()123 EX () EX () EX ()n n []L 1 =++++()μμμ μ n L 1 μμμ==()n .# X n

Question 9: Consider the following population of data: {140, 150, 160}. (i) Determine the mean and variance of the population. Total marks:2 1 μ =++==()140 150 160 450/ 3 150 3 11 1 σ 222=−=−[xnX ] [67700 ( 3 )( 22500 )] = [67700 −== 67500 ] 200 / 3 66 . 67 n ∑ i 3 3 11N ⎡ N ⎤ σμ22=−=()X X 2 − μ2 N ∑∑i N ⎢ i ⎥ i==1 ⎣ i 1 ⎦ (ii) Determine the sampling distribution of the sample mean for a sample of size 2. Graph this distribution with a simple bar graph. Total marks:2

X1,X2 X 140, 140 140 140, 150 145 140, 160 150 150, 140 145 150, 150 150 150, 160 155 160, 140 150 160, 150 155 160, 160 160

X P( X ) 140 1/9 145 2/9 150 3/9 155 2/9 160 1/9

P( X )

X 140 145 150 155 160

(iii) Determine the variance of X ? 66.67/2=33.335 Total marks:1

Question 10: Multiple Choice / True and False: Choose the best answer. (1 MARK EACH) MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) In a recent survey of high school students, it was found that the average amount of money spent 1) ______on entertainment each week was normally distributed with a mean of $52.30 and a standard deviation of $18.23. Assuming these values are representative of all high school students, what is the probability that for a sample of 25, the average amount spent by each student exceeds $60? A) 0.0174 B) 0.4826 C) 0.3372 D) 0.1628

2) If a sample of size 100 is taken from a population whose standard deviation is equal to 100, then 2) ______the standard error of the mean is equal to: A) 10 B) 1,000 C) 100 D) 10,000

3) What is the name of the parameter that determines the shape of the chi-square distribution? 3) ______A) mean B) variance C) degrees of freedom D) proportion

4) The sampling distribution of the mean is a distribution of: 4) ______A) individual population values. B) population parameters. C) sample statistics. D) individual sample values.

5) Why is the central limit theorem important in statistics? 5) ______A) Because for any population, it says the sampling distribution of the sample mean is approximately normal, regardless of the shape of the population. B) Because for a large sample size n, it says the population is approximately normal. C) Because for any sample size n, it says the sampling distribution of the sample mean is approximately normal. D) Because for a large sample size n, it says the sampling distribution of the sample mean is approximately normal, regardless of the shape of the population.

6) Which of the following distributions is used to determine the sampling distribution of the 6) ______sample variance? A) binomial distribution B) normal distribution C) chi-square distribution D) Poisson distribution

7) As the size of the sample increases, what happens to the shape of the sampling distribution of 7) ______sample means? A) It becomes positively skewed. B) It becomes uniformly distributed. C) It becomes negatively skewed. D) It becomes approximately normal.

8) Which of the following statements is true regarding the standard error of the mean? 8) ______A) It is equal to the population standard deviation divided by the square root of n. B) It is equal to the population variance divided by the square root of n. C) It is equal to the population standard deviation divided by the sample size n. D) It is equal to the population variance divided by (n -1).

9) The number of students using the ATM on campus daily is normally distributed with a mean of 9) ______237.6 and a standard deviation of 26.3. For a random sample of 55 days, what is the probability that the ATM usage averaged more than 230 students per day? A) 0.9483 B) 0.9838 C) 0.9524 D) 0.9756

10) The amount of time that you have to wait before seeing the doctor in the doctorʹs office is 10) ______normally distributed with a mean of 15.2 minutes and a standard deviation of 15.2 minutes. If you take a random sample of 35 patients, what is the probability that the average wait time is greater than 20 minutes? (Hint: Round the probability value to 2 decimal places.) A) 0.28 B) 0.03 C) 0.16 D) 0.09

TRUE/FALSE. Write ʹTʹ if the statement is true and ʹFʹ if the statement is false. 11) The central limit theorem states that as the sample size increases, the distribution of the 11) ______population mean approaches the normal distribution.

12) The chi-square family of distributions is used in applied statistical analysis because it provides a 12) ______link between the sample and population variances.

13) The central limit theorem is basic to the concept of because it permits us to 13) ______draw conclusions about the population based strictly on sample data.

14) If the sample size, n, equals the population size, N, then the variance of the sample mean, , is 14) ______zero.

15) The larger the sample size, the larger the standard error of the sample proportion. 15) ______

16) The central limit theorem states that the sampling distribution of sample means will closely 16) ______resemble the normal distribution regardless of the sample size.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 17) Based on the central limit theorem, the mean of all possible sample means is equal to the 17) ______population: A) variance. B) . C) mean. D) standard deviation.

TRUE/FALSE. Write ʹTʹ if the statement is true and ʹFʹ if the statement is false. 18) The standard error of the mean is also called sampling error. 18) ______

19) The variance of the sampling distribution of sample mean decreases as the sample size, n, 19) ______increases.

20) The mean and variance of a chi-square distribution with ν degrees of freedom is determined by 20) ______the number of degrees of freedom.

End of Exam

1) A 2) A 3) C 4) C 5) D 6) C 7) D 8) A 9) B 10) B 11) FALSE 12) TRUE 13) TRUE 14) TRUE 15) FALSE 16) FALSE 17) C 18) FALSE 19) TRUE 20)true TRUE