ECON10005 Lecture Notes - Lecture 8: Statistical Inference, Confidence Interval

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tealcattle862

29 Aug 2018

School

University of Melbourne

Department

Economics

Course

ECON10005

Professor

All

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Document Summary

Week (cid:1012) lecture (cid:1006: reject (cid:862)(cid:2868):=(cid:1865)(cid:863) for |(cid:1872)|=|(cid:4666) (cid:3040)(cid:4667) |>(cid:1872)2,(cid:3041) (cid:2869: do not reject (cid:862)(cid:2868):=(cid:1865)(cid:863) for |(cid:1872)|=|(cid:4666) (cid:3040)(cid:4667) |(cid:3409)(cid:1872)2,(cid:3041) (cid:2869: do not reject (cid:862)(cid:2868):=(cid:1865)(cid:863) for (cid:1872)2,(cid:3041) (cid:2869)(cid:3409)(cid:4666) (cid:3040)(cid:4667) (cid:1872)(cid:2870),(cid:3041) (cid:2869)(cid:3409)(cid:4666) (cid:1865)(cid:4667) (cid:3409)(cid:1872)(cid:2870),(cid:3041) (cid:2869) (cid:1871) (cid:1866: multiply both sides by (cid:3041) (cid:1872)(cid:2870),(cid:3041) (cid:2869) (cid:1871) (cid:1866)(cid:3409) (cid:1865)(cid:3409)(cid:1872)(cid:2870),(cid:3041) (cid:2869) (cid:1871) (cid:1866, minus both sides by (cid:1872)(cid:2870),(cid:3041) (cid:2869) (cid:1871) (cid:1866)(cid:3409) (cid:1865)(cid:3409) +(cid:1872)(cid:2870),(cid:3041) (cid:2869) (cid:1871) (cid:1866, multiply through by (cid:883) +(cid:1872)(cid:2870),(cid:3041) (cid:2869) (cid:1871) (cid:1866)(cid:3410)(cid:1865)(cid:3410) (cid:1872)(cid:2870),(cid:3041) (cid:2869) (cid:1871) (cid:1866: (cid:2868):=(cid:1865) is not rejected against : (cid:1865) for (cid:1865) in the interval. [ (cid:1872)(cid:2870),(cid:3041) (cid:2869) (cid:1871) (cid:1866): the probability that the interval includes is (cid:883) , definition. Confidence interval length: a short confidence interval implies the sample is informative about, a long confidence interval implies the sample is not so informative about ( +(cid:1872)(cid:2870),(cid:3041) (cid:2869) (cid:1871) (cid:1866)) ( (cid:1872)(cid:2870),(cid:3041) (cid:2869) (cid:1871) (cid:1866))=(cid:884)(cid:1872)(cid:2870),(cid:3041) (cid:2869) (cid:1871) (cid:1866, thus shorter (i. e. more informative) confidence intervals arise from.

Related Questions

1	Which of the following statements about Type I and Type II errors is correct
a	Type I: Reject a true alternative hypothesis. Type II: Do not reject a false alternative.
b	Type I: Reject a true null hypothesis. Type II: Do not reject a false null hypothesis.
c	Type I: Reject a false null hypothesis. Type II: Reject a true null hypothesis.
d	Type I: Do not reject a false null hypothesis. Type II: Reject a true null hypothesis.

2	You are reading a report that contains a hypothesis test you are interested in. The writer of the report writes that the p-value for the test you are interested in is 0.0749, but does not tell you the value of the test statistic. From this information you can:


a	Not reject the hypothesis at a Probability of Type I error = .05, but reject the hypothesis at a Probability of Type I error = 0.10

b	Reject the hypothesis at a Probability of Type I error = .05, and reject at a Probability of Type I error = 0.10

c	Not reject the hypothesis at a Probability of Type I error = 0.05, and not reject at a Probability of Type I error = 0.10

d	Reject the hypothesis at a Probability of Type I error = .05, but not reject at a Probability of Type I error = 0.10


3	The random sample below is obtained to test the following hypothesis about the population mean.
	H?: ? ?	1500
	H?: ? >	1500
	620	1711	366	2528	2678	1661	442	725	1938
	409	330	2480	542	369	2124	549	2074	1665
	1873	873	2143	2061	1177	2509	1264	2397	1523
	1837	1958	1041	1639	2199	2232	387	2270	2136
	1111	1883	2612	2230	1597	1726	694	1990	1354
	2090	909	2128	1608	747	1121	2220	2390	2347
	1041	316	655	632	2064	1901	532	552	846
	2704	1410	2165	1065	937	1452	2539	410	656
	1169	527	809	2364	2350	2210	1459	2391	856
	2711	1985	2382	2289	1927	518	2177	437	1151
	2018	1580	607	2715	2188	1691	1394	2610	1186
	695	2428	2246	858	2036	1681	2449	1578	1971
	1846	1729	2389	1737	1913	1863	2072	2593	2287
	2220	2230	551	458	2626	2731	488	2551	1736
	1373	307	1803	2647	2679	1508	1468	1443	516
	1002	2116	2616	817	2522	460	1879	1999	1837

	The level of significance of the test is ? = 0.05. Compute the relevant test statistic.
	This is a(n) _______ (two-tail, upper-tail, lower-tail) test. The test statistic is TS = _______.
a	Two-tail test		TS =	1.81
	Do not reject H?: ? ? 1500. Conclude that the population mean is not greater than 1500.
b	Upper tail test.		TS =	1.52
	Do not reject H?: ? ? 1500. Conclude that the population mean is not greater than 1500.
c	Upper tail test.		TS =	1.81
	Reject H?: ? ? 1500. Conclude that the population mean is greater than 1500.
d	Lower tail test.		TS =	1.98
	Do not reject H?: ? ? 1500. Conclude that the population mean is no greater than 1500.

4	Consider the following hypothesis test.
	H?: ? ?	30
	H?: ? >	30
	A random sample of n = 15 yielded the following observations
	51	38	26	16	28
	57	20	33	35	23
	21	47	56	54	36
	Use ? =	0.05
	TS = ______	CV = ______	State the decision rule.
a	1.68	1.761	Do not reject H?. Conclude the mean is not greater than 30.
b	1.68	1.64	Reject H?. Conclude the mean is greater than 30.
c	1.847	2.145	Do not reject H?. Conclude the mean is not less than 30.
d	1.847	1.761	Reject H?. Conclude the mean is less than 30.

5	In a recent study, a major fast food restaurant had a mean service time of 165 seconds. The company embarks on a quality improvement effort to reduce the service time and has developed improvements to the service process. The new process will be tested in a sample of stores. The new process will be adopted in all of its stores, if it reduced mean service time by more than 45 seconds compared to the current mean service time. To perform the hypothesis test, the sample of 48 stores yields the following data (seconds).





	90	96	133	108	136	110	119	138
	129	98	101	92	135	124	115	90
	132	125	110	124	126	138	94	130
	108	96	140	135	102	114	109	137
	138	104	108	134	92	107	96	119
	105	111	96	136	126	116	98	131

	Use ? =	0.05
	\|TS\| = ______	\|CV\| = ______
a	1.548	1.678	Do not reject H?. The mean service time is not reduced by more than 45 seconds. Do not adopt the new process.

b	1.871	1.678	Reject H?. The mean service time is reduced by more than 45 seconds. Adopt the new process.

c	1.871	1.640	Do not reject H?. The mean service time is not reduced by more than 45 seconds. Do not adopt the new process.

d	1.548	1.640	Reject H?. The mean service time is reduced by more than 45 seconds. Adopt the new process.

Individals are of two di Â§erent abilities: high (H) and low (L). An individual knows his or her own

ability but an employer does not. Employers believe that both types of individuals are equally likely

in the population, and therefore has an initial (prior) belief that an individual is high ability with

probability 1. Before an individual looks for a job, he or she has an opportunity to go to college. 2

Assume that attending college does not change an individualiÃs ability; this problem shows that even in

this circumstance, college can perform a useful social function. The e Â§ort cost of attending college is

di Â§erent for individuals of di Â§erent abilities. In order to attend college, high ability individuals must

expend e Â§ort 1, and low ability individuals must expend e Â§ort 1, where a < 3 (this means that a 3a

low ability individual has a higher cost of attending college). Not attending college costs 0. After

observing an individualiÃs education (that is, whether he or she attended college) an employer makes

the individual a job o Â§er. The wage that the individual receives is equal to the employeriÃs belief that

the individual is high ability, given the level of schooling that the individual has chosen to acquire. For

example, if the employer believes that a college graduate is high ability with probability 2 he will o Â§er 3

a college graduate a wage of 2 . If he believes that an individal with no college education is high ability 3

with probability 1 , he will o Â§er an individual without college a wage of 1 : An individualiÃs payo Â§ is 44

equal to his wage, minus the cost of education..

First consider a separating equilibrium. In a separating equilibrium, high ability individuals attend college, but low ability individuals do not.

(a) If the employer observes that an individual did not attend college, what is his belief about the individualiÃs ability? What wage will he o Â§er individuals who do not attend college?
(b) If the employer observes that an individual did attend college, what is his belief about the indi- vidualiÃs ability? What wage will he o Â§er individuals who attend college?
(c) Given the wages paid to college graduates and those who do not attend, will the high-ability individual chooose to attend college? (Hint: compare the payo Â§ of attending college for the high ability graduate to the payo Â§ of not attending college)
(d) Given the wages paid to college graduates and those who do not attend, for which values of a will the low-ability individual chooose to attend college? (Hint: compare the payo Â§ of attending college for the low ability graduate to the payo Â§ of not attending college)
(e) In a separating equilibrium, a low ability individual does not attend college. For which values of a is this expectation consistent with the low ability individualiÃs strategy?

Now, consider a pooling equilibrium in which all individuals attend college; that is, both high and low ability individuals attend college. If some individual were to choose not to attend college, the employer would believe that he or she must be low ability, and would o Â§er a wage of 0.
(f) If the employer observes that an individual attended college, what is his belief about the individ- ualiÃs ability? What wage will he o Â§er to college graduates? (Hint: The employer expects that all individuals attend college. This means that attending college conveys no information about the individualiÃs true ability).
(g) Given the wages paid to college graduates (and the wage that would be paid to non-grads) would the high-ability individual choose to attend college? For which values of a would the low ability individual choose to attend college?
(h) In a pooling equilibrium, both high and low ability individuals attend college. For which values of a is this expectation consistent with the individualsiÃstrategies?

Finally, consider a sem-separating equilibrium in which high-ability individuals attend college (for sure), but low ability students attend college with probability 0 < s < 1:
(i) If the employer sees that an individual attends college what is his belief that the individual is high-ability? What is the wage to college graduates? (Hint: Remember, all high ability individuals attend, but low ability individuals attend with probability s. Divide the probability of an individual being high ability and attending college by the overall probability of an individual attending college).

(j) If the employer sees that an individual did not attend college, what are his beliefs about the individualiÃs ability? (Hint: all high ability individuals attend college)
(k) The low-ability individual is willing to randomize between attending college and not attending college. Use this to OÃËnd the probability that the low-ability individual attends in equilibrium.

Bonus: Comment on the purpose of college illustrated in this example. How is this purpose a Â§ected as college becomes easier for the low-ability type (i.e. as a increases)?

lavendergnat675

Question 1

Which of the following approaches to understanding and predicting consumer behavior depends primarily on the knowledge and experience of a firm's employees and its suppliers?

		Test marketing and price experiments
		Analysis of historical data.
		Direct consumer surveys.
		Expert opinion.

Question 2

In which of the following situations would reliance on expert opinion as a basis for a managerial decision be most preferred?

		When the product being marketed is relatively new.
		When the level of economic activity can have a significant effect on the demand for the firm's output.
		When the product can be packaged with a variety of price and quality combinations.
		When the business in question serves as a supplier of inputs to other businesses, especially in multi-product situations where other strategies may be prohibitively expensive.

Question 3

The approach to analyzing consumer behavior that asks consumers to rank and choose among different product attributes to reveal their relative valuation of different characteristics is called:

		conjoint analysis.
		contingent valuation.
		the hedonic estimation technique.
		a direct consumer survey.

Question 4

All of the following are limitations of direct consumer surveys except:

		the possibility of response biases because survey respondents may not want to reveal their true preferences.
		the possibility that the type of questions asked may unintentionally bias the respondent's answers.
		the likelihood that respondents will deliberately and systematically mislead interviewers.
		the possibility that consumers' responses may not reflect their actual behavior in the market place.

Question 5

Which of the following approaches to understanding and predicting consumer behavior does not actually solicit any information from potential customers?

		Expert opinion.
		Test marketing.
		Analysis of historical data.
		Conjoint analysis.

Question 6

Data collected on a sample of individuals with different characteristics at a specific point in time are called:

		panel data.
		cross-section data.
		time series data.
		none of the above.

Question 7

Which of the following approaches to understanding and predicting consumer behavior provides the most insight into how consumers can be expected to respond in an actual market setting?

		Test marketing.
		Conjoint analysis.
		Expert opinion.
		Analysis of historical data.

Question 8

An approach to analyzing consumer behavior in which consumer reaction to different prices is analyzed in a laboratory situation or a test market is called:

		non-price experiments.
		focus groups.
		price experiments.
		none of the above.

Question 9

Data collected on the same observation unit at a number of points in time are called:

		panel data.
		time series data.
		cross-section data.
		none of the above.

Question 10

A measure of how much the coefficient would vary in regressions based on different samples is called:

		F-statistic.
		standard error of the estimated coefficient.
		t-statistic.
		partial F-statistic.

Question 11

The test statistic used to test the hypothesis of whether a regression coefficient is significantly different from zero, holding all other independent variables constant, is called a(n):

		t-test.
		F-test.
		multicollinearity test.
		autocorrelation test.

Question 12

Regression analysis that analyzes the relationship between one dependent variable and several independent variables is called:

		cluster analysis.
		correlation analysis.
		multiple regression analysis.
		simple regression analysis.

Question 13

The ratio of the regression coefficient to its standard error is called:

		F-statistic.
		t-statistic.
		coefficient of determination.
		partial F-statistic.

Question 14

The coefficient of determination will range between what values?

		-1 and +1
		0 and 1
		-3 and +3
		none of the above

Question 15

The range of values in which we can be confident that the true regression coefficient lies within a given degree of probability is called a:

		confidence interval.
		logistic regression.
		prediction interval.
		none of the above.

Question 16

The estimated regression equation is Y = 10 + 2.5X, if X =0 than the predicted value of Y is equal to

		2.5
		7.5
		12.5
		10

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ECON10005 Lecture Notes - Lecture 8: Statistical Inference, Confidence Interval

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