代写MGT403: Probability & Statistics Fall 2023帮做R程序

日期：2024-09-27 06:14

MGT403: Probability?&?Statistics

Fall 2023

Statistics Examination - 2023

Analytic Part

Question 1 (29 points)

The US Fisheries and?Wildlife Service?(FWS)?requires?that?the?average?meat?per?mussel?in?any?ship’s mussel?catch must be?at least?0.5?ounces.

Suppose?the FWS draws?a?simple?random?sample?of?1,000?mussels?from?a?ship?with?a?large?mussel catch.?The average meat?per?mussel?in?the?sample?is?0.48?ounces,?and?the?sample???standard?deviation is?0.3?ounces.

You may use?the Normal Distribution approximation?for?all?calculations in?this problem?because?the?sample?size is?sufficiently large.

a.??(4 points).?What is?the standard?error?for?the?estimated?average meat per mussel,?given?the?sample size?and?the?estimated?standard?deviation?

b. ?(4 points). Find a 90% confidence interval?for?the average meat per mussel?in?the?ship’s?entire?catch. Interpret?this confidence interval?(in?precise?words).

c.??(4 points). Suppose?that?the?true?average?meat per mussel?in?the?ship’s?entire?catch?is?0.5??ounces, and?the standard?deviation?of?the meat?per?mussel?in?the?ship’s?entire?catch?is?0.3?ounces.?What is?the probability?that?the average meat?per?mussel?in?a?simple?random sample of 1,000?mussels?will be?0.48?ounces?or less?

The?FWS enforces its mussel meat requirement?by?imposing penalties based?on?the?results?of?its random sample of 1,000 mussels?from?each?ship’s?catch.?The penalties?for?under-weight?catches?are?as?follows:

. ???If?the average meat per mussel is?greater?than?0.5 ounces,?then no penalty is imposed.

. ???If?the average meat per mussel in?the sample is less?than?0.45?ounces,?then?the FWS?confiscates?the ship’s?entire?catch.

. ???If?the average meat per mussel is between?0.45 and?0.5?ounces,?then?the FWS?confiscates a?fraction of?the?ship’s?entire?catch?according?to?the?formula:

Fraction?confiscated?=?20?×(0.5?一?average?meat?per?mussel?in?the?sample)

d. ?(4 points) Suppose, as in part (c),?that?the?true average meat per mussel?in?a?ship’s?entire

catch is?0.5 ounces,?and?the?standard?deviation?of?the meat per mussel?in?the?ship’s?entire catch is?0.3?ounces. Find?the?5th?and?the?95th?percentile of?the probability?distribution?for?the?fraction of?the?catch?that?will be?confiscated by?the FWS.

The FWS believes?that its current penalty?scheme?is?too?arbitrary?since?its?penalties?are based on a sample?and hence?are?subject?to?sampling?error.?It?has?also?received?a?complaint?from?the?operator of?the above?ship?who?asserts?that?their?catch?complies?with regulations?even?though?the average meat per mussel?in?the?sample?falls?short?of?the?FWS’?threshold.

Consider?the above sample?of 1,000?mussels?with?an?average?meat?per?mussel?of?0.48?ounces and a sample?standard?deviation?of?0.3?ounces.?The FWS?would like?to?assess?the?operator’s claim?that?the average meat per mussel in?the?ship’s?entire?catch?exceeds?the?FWS’?threshold?of 0.5 ounces:?the sample results?reflect?pure?chance.?They?perform.?a hypothesis?test?at?a?significance level?of?α = 1% using?the?sample data.

e.??(2 points)?State?the null hypothesis and?alternative?hypothesis?for?this?test.

f. ??(4 points) Draw a?graph?that shows?the set-up?to?test?the null hypothesis. Place?the?sample?average on?your?graph?and use it?to?indicate?the p-value?for?your?test?in?the?graph. Label?all?relevant elements?of?your?graph.

g.??(3 points) Calculate?the?test statistic?and?p-value?for?your?test. Based?on?your?computation,?do?you reject or?fail?to reject?the?null?hypothesis?

h. ?(4 points)?The FWS decides?that?they are comfortable?with?their current penalty rule if?they?can reject?the?test’s null hypothesis at?a 1%?significance level?for?samples like?this?one.?They?wonder how large a?sample?they?would need?to?collect?to?ensure?this?is?the?case.

Given?the estimated standard?deviation?of?0.3?ounces?and?average?of?0.48?ounces?of?meat,??what is?the range of?values?for?sample?size N for?which?you?would reject?the null hypothesis?at?the 1%?significance level?

Question 2 (16 points)

A consultant is?working?on?a project?for?a?client?who?would like?to understand?how?to?allocate?advertising budgets across?social media platforms?to?drive?sales.

The client provides?the consultant?with?three?years’?worth?of?data?on?weekly?units?sold?(in??thousands) and?the?fraction of?the?firm’s?weekly?advertising budget?spent?on?each?of?three?social media platforms: Facebook, Instagram, and?YouTube.?Thus, if?the?client?spent?$1?on??Facebook out of?a budget?of?$10,?they?would?say?the?fraction?spent?on Facebook?is?0.10.

The data show?the?following?descriptive patterns:

	Average	SD
Sales (000)	14.13	5.26
Fraction Spent, Facebook	0.15	0.14
Fraction Spent, Instagram	0.66	0.22

Thus,?the client spent,?on?average,?a?frac?on?of?0.19?of?their?adver?sing budget on?YouTube.

The consultant also performs?a?correla?on?analysis.?Each?entry?in?the?following?table corresponds?to?the correla?on between?the corresponding row?and?column?variables,?e.g.,?the?correla?on between?the?frac?on?of?the?weekly budget?spent?on Facebook?and?weekly?sales in??units?is?-0.31.

Sales (000)		Fraction Spent
		Facebook	Instagram
Sales (000) Fraction Spent, Facebook Fraction Spent, Instagram	1.00 -0.31 0.48	1.00 -0.78	1.00

The consultant runs a regression?of?sales?(left hand?side)?on?fraction?spent?on Instagram?(right?hand?side).

a.??(3 points).?What is?the coefficient?on?fraction?spent?on Instagram?for?this model?

b. ?(3 points).?What is?the intercept?for?this model?

c.??(2 points).?What is?the?model’s R-square?

After running?this regression,?the consultant decides?that?it?is?easier?to?think?in percentages?than?fractions. She?therefore expresses?spending?on Instagram?as?a percentage?of?the?total?budget.

That is, suppose?now:

percentage, Instagram?=?100 × Total Budget/Instagram Budget

The consultant?then runs?a new regression?of?sales?(left hand?side)?on percentage?spent?on?Instagram (right?hand?side).

Answer?the questions in parts (d)?through?(g),?providing?a?one-sentence?explanation?for?your?answer.

d. ?(2 points). How?would?the slope of?this new model change relative?to?the slope?you?estimated in?part?(a)?

e.??(2 points) How?would?the intercept change relative?to?the?intercept?you?estimated?in?part?(b)?

f. ??(2 points) How?would?the regression?standard error change relative?to?the regression?standard error?from?the model using?the?fraction?spent?on Instagram?

The consultant also?worries?that?sales?do not reflect?only Instagram?advertising?in?isolation, but?also advertising on?the?other?social media platforms.?She?therefore?considers?estimating?the following model:

sales?=?a?+ b1 ?× Fraction, Instagram?+ b2 ?× Fraction,?Facebook?+ e

g.??(2 points) Using?the information?from?this problem, determine how?the?inclusion?of?the?variable?fraction?spent on Facebook?would affect?the?coefficient?on?fraction?spent?on

Instagram, relative?to?the coefficient?in?part?(a).

The coefficient on?Fraction?spent?on?Instagram?(check?one?answer?only):

o?Increases? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ??Decreases

o?Remains?the?same ????????????????????????????????o?Too little information?to?tell

Provide a?concise?explanation?for?your?choice.

Question 3 (18 points)

Steve has?just accepted?a?job in?San Francisco?and?will?soon put?his?house in New Haven?up?for??sale. However, before purchasing a new home?in?the Bay?Area,?he?wants?to?get?an?idea?of?what price he can?expect?to?get?for his New Haven house. He?is?also?considering?what, if?any,?renovations might be?worthwhile before?selling?the house.

Steve has obtained?the prices?of?50 homes?that?were?sold?in New Haven?County in?the?past?month.?The data?set includes?the?following?variables:

Price: ???????????????Actual selling price of?the house in?thousands of dollars.

Square Footage:??Size of?the home’s living?area?in?square?feet.

Bedrooms: ?????????Number of bedrooms?in?the?house.

Bathrooms:????????Number of bathrooms?(“half”?baths?count?as?0.5).

Garage Size: ???????Number of?cars?that can park in?the?home’s?garage.

He runs a regression?of Price?on?the?other?variables?and?obtains?the?following results:

? ? ??

a.??(5 points) Steve’s house?is 1900?square?feet, has?three bedrooms,?one?and?a?half

bathrooms, and a?small?one-car?garage. Based?on?the?regression?model?above,?what?is?the predicted?selling price?for his?house?

b. ?(5 points) Steve is considering renovating the?front half-bathroom before selling?the

house.?Specifically, he?could break?down?the?exterior?wall?to his half-bath?and?expand?it?to a?full-sized bathroom.?This?will add?20?square?feet?to?the?house.?The?project?will?cost?$15,000.?What is?Steve’s expected?net?profit?from?this?plan?

c.??(4 points)?Alternatively,?Steve could?spend?$10,000?to remodel his?garage?to

accommodate?two cars instead?of?one.?This plan?will leave?the?home’s living?area?and?other?features unchanged.?What is?Steve’s expected?net?profit?from?this?plan?

d.??(4 points) Because Steve has?only?50?observations?in his?regression, he?is?concerned?that?sampling error may?cause him?to make?the?wrong?decision?with respect?to?the?garage

remodel. Does?the regression provide evidence?at?the?5% level?that?Steve?should?expect?to make a positive net profit?on?the?garage remodel??(I.e.,?can?you reject?the hypothesis?that he should expect?to?make?a?negative?profit?)?Provide?the?test-statistic, p-value,?and?conclusion?for?this?test.

Even?though?Steve has only 50?observations,?assume?for?the purposes?of?this problem?that?you can use?the Normal?distribution?to?calculate?the p-value?for?this?test.

e.??(3 points extra credit)?Steve?believes?that?the?effect?of?the?number?of?bedrooms?on?sale?price is nonlinear. Specifically, he?believes?that?the?sale?price?increases?for?each additional bedroom at a?constant?amount?for?the?first,?second,?and?third bedroom. He believes?that?the sale price increases by?a?smaller?constant?amount?for?the?fourth?and fifth bedroom, but?that any?additional?bedrooms beyond?five?do?not?generate?any?value.

Steve?would like?to construct one?or?multiple?variables?that?would?allow?him?to?test his?beliefs about?the impact?of?the number?of?bedrooms?on?sale price (in?addition?to?the remaining?variables he controls?for?–?square?footage, bathrooms,?and?garage?size).

Define?the appropriate?variable?or?variables; be?clear?and precise.